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< div class= "chlinktop">< span class= "chlink1">Goto Chapter: </ span><a href= "chap0_mj.html">Top< /a> <a href="chap1_mj.html">1</a> <a href="chap2_mj.html">2</a> <a href="chap3_mj.html">3</a> <a href="chap4_mj.html">4</a> <a href="chap5_mj.html">5</a> <a href="chap6_mj.html">6</a> <a href="chapInd_mj.html">Ind</a> </div>
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<div class="ChapSects"><a href="chap1_mj.html#X808DC49C7ED99B52">1 <span class="Heading">Monoidal Categories</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap1_mj.html#X808DC49C7ED99B52">1.1 <span class="Heading">Monoidal Categories</span></a>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X84997A1E8188D6BE">1.1-1 TensorProductOnMorphisms</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X85008BF07C2C0386">1.1-2 TensorProductOnMorphismsWithGivenTensorProducts</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X84C2FAEA7B9678AA">1.1-3 AssociatorRightToLeft</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X85645C4D86060ED2">1.1-4 AssociatorRightToLeftWithGivenTensorProducts</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X7C5A064C7E1995F1">1.1-5 AssociatorLeftToRight</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X7C59D73F8389E389">1.1-6 AssociatorLeftToRightWithGivenTensorProducts</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X7EBCAFDD81BDB655">1.1-7 LeftUnitor</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X7ED284DF7BA21774">1.1-8 LeftUnitorWithGivenTensorProduct</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X79A274078147A9F7">1.1-9 LeftUnitorInverse</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X849609447CAE996B">1.1-10 LeftUnitorInverseWithGivenTensorProduct</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X82F758CB7A23468A">1.1-11 RightUnitor</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X84C8F97F7BF2282D">1.1-12 RightUnitorWithGivenTensorProduct</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X82DC825480E87301">1.1-13 RightUnitorInverse</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X85A8A87C87B33008">1.1-14 RightUnitorInverseWithGivenTensorProduct</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X797034F17AA47EE0">1.1-15 TensorProductOnObjects</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X7BF7FFDF8789474A">1.1-16 TensorUnit</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap1_mj.html#X7FB8551A815BEC53">1.2 <span class="Heading">Additive Monoidal Categories</span></a>
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<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X7ACF268E7D82B0C9">1.2-1 LeftDistributivityExpanding</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X7CA18968862D2285">1.2-2 LeftDistributivityExpandingWithGivenObjects</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X780322FE7E5A626D">1.2-3 LeftDistributivityFactoring</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X784D8CD685F5F021">1.2-4 LeftDistributivityFactoringWithGivenObjects</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X8409C74778A4313A">1.2-5 RightDistributivityExpanding</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X7DBE629F811DEADC">1.2-6 RightDistributivityExpandingWithGivenObjects</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X86C5C33786C1DC28">1.2-7 RightDistributivityFactoring</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X795267217F7807CE">1.2-8 RightDistributivityFactoringWithGivenObjects</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap1_mj.html#X7D94AD967E6D60F1">1.3 <span class="Heading">Braided Monoidal Categories</span></a>
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<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X7C6ADBFE7A8DD1E3">1.3-1 Braiding</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X7F4B09A87B2500C0">1.3-2 BraidingWithGivenTensorProducts</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X8195EC5279D3E8D0">1.3-3 BraidingInverse</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X854AFBC67C836769">1.3-4 BraidingInverseWithGivenTensorProducts</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap1_mj.html#X85ED71067F7CEA82">1.4 <span class="Heading">Symmetric Monoidal Categories</span></a>
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</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap1_mj.html#X85035E9683B050D0">1.5 <span class="Heading">Left Closed Monoidal Categories</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X87A4C0637EB8A18E">1.5-1 LeftInternalHomOnObjects</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X8439FF407D2C9C6E">1.5-2 LeftInternalHomOnMorphisms</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X7861C2AD82F62C77">1.5-3 LeftInternalHomOnMorphismsWithGivenLeftInternalHoms</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X82C4FACF83C3611D">1.5-4 LeftClosedMonoidalEvaluationMorphism</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X8575A3748713EACF">1.5-5 LeftClosedMonoidalEvaluationMorphismWithGivenSource</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X7A526C59812F2A59">1.5-6 LeftClosedMonoidalCoevaluationMorphism</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X7BC7F1497A00E87A">1.5-7 LeftClosedMonoidalCoevaluationMorphismWithGivenRange</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X83A5A47979C9B92D">1.5-8 TensorProductToLeftInternalHomAdjunctMorphism</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X7BB734AA7F213B47">1.5-9 TensorProductToLeftInternalHomAdjunctMorphismWithGivenLeftInternalHom</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X7C7D769879A7DA90">1.5-10 LeftInternalHomToTensorProductAdjunctMorphism</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X80431C73861AEC43">1.5-11 LeftInternalHomToTensorProductAdjunctMorphismWithGivenTensorProduct</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X79AC308B847F5177">1.5-12 LeftClosedMonoidalPreComposeMorphism</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X78E3E67480AC81F4">1.5-13 LeftClosedMonoidalPreComposeMorphismWithGivenObjects</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X7CFBE32D7F8F16CD">1.5-14 LeftClosedMonoidalPostComposeMorphism</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X7D5E05257D908A56">1.5-15 LeftClosedMonoidalPostComposeMorphismWithGivenObjects</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X859959ED7F2CFDCC">1.5-16 LeftDualOnObjects</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X7B697FFC7B23A48B">1.5-17 LeftDualOnMorphisms</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X8282E454874316D8">1.5-18 LeftDualOnMorphismsWithGivenLeftDuals</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X7FF3227087ABDA8E">1.5-19 LeftClosedMonoidalEvaluationForLeftDual</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X86995E4D7A63E68F">1.5-20 LeftClosedMonoidalEvaluationForLeftDualWithGivenTensorProduct</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X85D2B94A85E3DBD5">1.5-21 MorphismToLeftBidual</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X868774B68189D7E9">1.5-22 MorphismToLeftBidualWithGivenLeftBidual</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X7939CEE8854017D2">1.5-23 TensorProductLeftInternalHomCompatibilityMorphism</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X798C160A84CA5623">1.5-24 TensorProductLeftInternalHomCompatibilityMorphismWithGivenObjects</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X8635D47E7835C8AA">1.5-25 TensorProductLeftDualityCompatibilityMorphism</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X8435799E83221130">1.5-26 TensorProductLeftDualityCompatibilityMorphismWithGivenObjects</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X80FE157E81B02827">1.5-27 MorphismFromTensorProductToLeftInternalHom</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X78FDDC2986E50825">1.5-28 MorphismFromTensorProductToLeftInternalHomWithGivenObjects</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X7EA425DA848670D9">1.5-29 IsomorphismFromLeftDualObjectToLeftInternalHomIntoTensorUnit</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X781A2D417A8BC8DF">1.5-30 IsomorphismFromLeftInternalHomIntoTensorUnitToLeftDualObject</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X82D9D78C7A8DA61A">1.5-31 UniversalPropertyOfLeftDual</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X7C7A252A78484873">1.5-32 LeftClosedMonoidalLambdaIntroduction</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X83B3A8FA8363F620">1.5-33 LeftClosedMonoidalLambdaElimination</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X7ECAB43778388C41">1.5-34 IsomorphismFromObjectToLeftInternalHom</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X874D7D8E81A6AFF1">1.5-35 IsomorphismFromObjectToLeftInternalHomWithGivenLeftInternalHom</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X7908CF267DDAA38C">1.5-36 IsomorphismFromLeftInternalHomToObject</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X7B2F235C8444803C">1.5-37 IsomorphismFromLeftInternalHomToObjectWithGivenLeftInternalHom</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap1_mj.html#X7BC682067A30E580">1.6 <span class="Heading">Closed Monoidal Categories</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X78AA83E77B380D68">1.6-1 InternalHomOnObjects</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X7C0191A483A72F98">1.6-2 InternalHomOnMorphisms</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X861226B47B5CB713">1.6-3 InternalHomOnMorphismsWithGivenInternalHoms</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X844F05DF7DE23F99">1.6-4 ClosedMonoidalRightEvaluationMorphism</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X83E6461280B0EB0C">1.6-5 ClosedMonoidalRightEvaluationMorphismWithGivenSource</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X7A745E5A836B5E91">1.6-6 ClosedMonoidalRightCoevaluationMorphism</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X868AD1A278EBAB03">1.6-7 ClosedMonoidalRightCoevaluationMorphismWithGivenRange</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X7BA9C4A184346304">1.6-8 TensorProductToInternalHomRightAdjunctMorphism</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X8218B01287CD44E5">1.6-9 TensorProductToInternalHomRightAdjunctMorphismWithGivenInternalHom</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X78A16762823E5F1C">1.6-10 TensorProductToInternalHomRightAdjunctionIsomorphism</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X81D37FA282FC8F11">1.6-11 TensorProductToInternalHomRightAdjunctionIsomorphismWithGivenObjects</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X8271B2767A2DFE76">1.6-12 InternalHomToTensorProductRightAdjunctMorphism</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X7ACBB0CE7FA4C781">1.6-13 InternalHomToTensorProductRightAdjunctMorphismWithGivenTensorProduct</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X79EEE4757C27C26E">1.6-14 InternalHomToTensorProductRightAdjunctionIsomorphism</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X7B4CD74A7CE51263">1.6-15 InternalHomToTensorProductRightAdjunctionIsomorphismWithGivenObjects</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X80B3C53A854B515B">1.6-16 ClosedMonoidalLeftEvaluationMorphism</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X84E1E136819BDA89">1.6-17 ClosedMonoidalLeftEvaluationMorphismWithGivenSource</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X7E35DEB787A71A1F">1.6-18 ClosedMonoidalLeftCoevaluationMorphism</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X8196CE5D7C88D83C">1.6-19 ClosedMonoidalLeftCoevaluationMorphismWithGivenRange</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X80A58B1F80A186BD">1.6-20 TensorProductToInternalHomLeftAdjunctMorphism</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X7CCE373482D60A57">1.6-21 TensorProductToInternalHomLeftAdjunctMorphismWithGivenInternalHom</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X793BAA50819573E4">1.6-22 TensorProductToInternalHomLeftAdjunctionIsomorphism</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X791276987824CDF8">1.6-23 TensorProductToInternalHomLeftAdjunctionIsomorphismWithGivenObjects</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X85F426677EB81BCF">1.6-24 InternalHomToTensorProductLeftAdjunctMorphism</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X859B738881052D1C">1.6-25 InternalHomToTensorProductLeftAdjunctMorphismWithGivenTensorProduct</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X86AD79F17F8CEE96">1.6-26 InternalHomToTensorProductLeftAdjunctionIsomorphism</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X7D707B26863D508A">1.6-27 InternalHomToTensorProductLeftAdjunctionIsomorphismWithGivenObjects</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X86E4E97C82BFB45C">1.6-28 MonoidalPreComposeMorphism</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X7FE4A52284060F20">1.6-29 MonoidalPreComposeMorphismWithGivenObjects</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X79F4FCE781385829">1.6-30 MonoidalPostComposeMorphism</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X7E6933B8816EE4E3">1.6-31 MonoidalPostComposeMorphismWithGivenObjects</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X80FFC71D7E57DD53">1.6-32 DualOnObjects</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X815C0BC47D000819">1.6-33 DualOnMorphisms</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X7DCC0F468386AA46">1.6-34 DualOnMorphismsWithGivenDuals</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X86D42C7587F03A68">1.6-35 EvaluationForDual</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X7B588B1B8472834E">1.6-36 EvaluationForDualWithGivenTensorProduct</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X7F68B55781C1DFB1">1.6-37 MorphismToBidual</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X79260E5680F1E741">1.6-38 MorphismToBidualWithGivenBidual</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X80CDA0CB821E08EE">1.6-39 TensorProductInternalHomCompatibilityMorphism</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X7B0194A07AB73486">1.6-40 TensorProductInternalHomCompatibilityMorphismWithGivenObjects</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X7D173DB08132E40A">1.6-41 TensorProductDualityCompatibilityMorphism</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X794ED97782B8DF76">1.6-42 TensorProductDualityCompatibilityMorphismWithGivenObjects</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X8439BFF57BFE390F">1.6-43 MorphismFromTensorProductToInternalHom</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X8473D28B7F3A5E20">1.6-44 MorphismFromTensorProductToInternalHomWithGivenObjects</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X8430726C82D6CFF7">1.6-45 IsomorphismFromDualObjectToInternalHomIntoTensorUnit</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X7A015736812A97C6">1.6-46 IsomorphismFromInternalHomIntoTensorUnitToDualObject</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X7AF8F68887146C20">1.6-47 UniversalPropertyOfDual</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X80F3F2287B9E55E3">1.6-48 LambdaIntroduction</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X79EC44077F661E80">1.6-49 LambdaElimination</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X8726BA888787D53D">1.6-50 IsomorphismFromObjectToInternalHom</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X87F91EF37C76A7F7">1.6-51 IsomorphismFromObjectToInternalHomWithGivenInternalHom</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X831CF4E1824FA178">1.6-52 IsomorphismFromInternalHomToObject</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X7988DC7C79BED3B2">1.6-53 IsomorphismFromInternalHomToObjectWithGivenInternalHom</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap1_mj.html#X86546D277A535EB1">1.7 <span class="Heading">Left Coclosed Monoidal Categories</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X7CE5D10B7F10E9CC">1.7-1 LeftInternalCoHomOnObjects</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X853213767BB48099">1.7-2 LeftInternalCoHomOnMorphisms</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X7F302E3880752A2F">1.7-3 LeftInternalCoHomOnMorphismsWithGivenLeftInternalCoHoms</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X819A762D83B1C294">1.7-4 LeftCoclosedMonoidalEvaluationMorphism</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X7E5607B1789E00B7">1.7-5 LeftCoclosedMonoidalEvaluationMorphismWithGivenRange</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X81A249CE84828A9D">1.7-6 LeftCoclosedMonoidalCoevaluationMorphism</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X7D1018A284F21140">1.7-7 LeftCoclosedMonoidalCoevaluationMorphismWithGivenSource</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X851216B2807B3900">1.7-8 TensorProductToLeftInternalCoHomAdjunctMorphism</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X87BFA0FA7AFF2868">1.7-9 TensorProductToLeftInternalCoHomAdjunctMorphismWithGivenLeftInternalCoHom</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X7DDDACD58454C3A4">1.7-10 LeftInternalCoHomToTensorProductAdjunctMorphism</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X821AC96981CA794C">1.7-11 LeftInternalCoHomToTensorProductAdjunctMorphismWithGivenTensorProduct</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X8416357982DCBEE4">1.7-12 LeftCoclosedMonoidalPreCoComposeMorphism</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X7EECDE4583E14A59">1.7-13 LeftCoclosedMonoidalPreCoComposeMorphismWithGivenObjects</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X80A270E982F5BC2D">1.7-14 LeftCoclosedMonoidalPostCoComposeMorphism</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X867C6F087D72AB3F">1.7-15 LeftCoclosedMonoidalPostCoComposeMorphismWithGivenObjects</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X85A63AA57F0678EE">1.7-16 LeftCoDualOnObjects</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X7949DE307D2D5083">1.7-17 LeftCoDualOnMorphisms</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X84C835CB7E708596">1.7-18 LeftCoDualOnMorphismsWithGivenLeftCoDuals</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X7A20AC6A837F79BA">1.7-19 LeftCoclosedMonoidalEvaluationForLeftCoDual</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X87B15AC678955783">1.7-20 LeftCoclosedMonoidalEvaluationForLeftCoDualWithGivenTensorProduct</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X7F67676A85A47BDB">1.7-21 MorphismFromLeftCoBidual</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X7AF5F77180E09AAE">1.7-22 MorphismFromLeftCoBidualWithGivenLeftCoBidual</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X84331A1B7F62782F">1.7-23 LeftInternalCoHomTensorProductCompatibilityMorphism</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X808BEB0879D58CAF">1.7-24 LeftInternalCoHomTensorProductCompatibilityMorphismWithGivenObjects</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X7B4A66A8803C99BA">1.7-25 LeftCoDualityTensorProductCompatibilityMorphism</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X86E72092788A8F2D">1.7-26 LeftCoDualityTensorProductCompatibilityMorphismWithGivenObjects</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X7A5FDE4B83C5F328">1.7-27 MorphismFromLeftInternalCoHomToTensorProduct</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X7BD2A33F84A2FBAB">1.7-28 MorphismFromLeftInternalCoHomToTensorProductWithGivenObjects</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X7CC10C9E7EA7EED4">1.7-29 IsomorphismFromLeftCoDualObjectToLeftInternalCoHomFromTensorUnit</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X87F96B047F320A17">1.7-30 IsomorphismFromLeftInternalCoHomFromTensorUnitToLeftCoDualObject</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X7A0DE1667FF28457">1.7-31 UniversalPropertyOfLeftCoDual</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X85960F7479FFA8DA">1.7-32 LeftCoclosedMonoidalLambdaIntroduction</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X8721DDEC7DF9D89D">1.7-33 LeftCoclosedMonoidalLambdaElimination</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X853342457CC3B489">1.7-34 IsomorphismFromObjectToLeftInternalCoHom</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X82709BED807CB6FC">1.7-35 IsomorphismFromObjectToLeftInternalCoHomWithGivenLeftInternalCoHom</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X806C843B82108148">1.7-36 IsomorphismFromLeftInternalCoHomToObject</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X8370164B7EAF833D">1.7-37 IsomorphismFromLeftInternalCoHomToObjectWithGivenLeftInternalCoHom</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap1_mj.html#X8197D69B805349AC">1.8 <span class="Heading">Coclosed Monoidal Categories</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X791BE0B0847A430D">1.8-1 InternalCoHomOnObjects</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X7FDF105D7F56EB53">1.8-2 InternalCoHomOnMorphisms</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X84DDA3DD878162B9">1.8-3 InternalCoHomOnMorphismsWithGivenInternalCoHoms</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X83CEC88D79AE5684">1.8-4 CoclosedMonoidalRightEvaluationMorphism</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X79333EC9822EA316">1.8-5 CoclosedMonoidalRightEvaluationMorphismWithGivenRange</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X80CA11547F3909E3">1.8-6 CoclosedMonoidalRightCoevaluationMorphism</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X7F1C10897EC4D14E">1.8-7 CoclosedMonoidalRightCoevaluationMorphismWithGivenSource</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X82B0937678F04730">1.8-8 TensorProductToInternalCoHomRightAdjunctMorphism</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X833D7B567810D7F3">1.8-9 TensorProductToInternalCoHomRightAdjunctMorphismWithGivenInternalCoHom</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X8623CC567D687FFD">1.8-10 InternalCoHomToTensorProductRightAdjunctMorphism</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X821B5212835BD1F6">1.8-11 InternalCoHomToTensorProductRightAdjunctMorphismWithGivenTensorProduct</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X81020D9E85C3280A">1.8-12 CoclosedMonoidalLeftEvaluationMorphism</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X7A1801D77EECEA29">1.8-13 CoclosedMonoidalLeftEvaluationMorphismWithGivenRange</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X7D6905F582F06003">1.8-14 CoclosedMonoidalLeftCoevaluationMorphism</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X802A3FC48280FBDE">1.8-15 CoclosedMonoidalLeftCoevaluationMorphismWithGivenSource</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X7FB4953B7F813D0B">1.8-16 TensorProductToInternalCoHomLeftAdjunctMorphism</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X8312DB0A82A97D4B">1.8-17 TensorProductToInternalCoHomLeftAdjunctMorphismWithGivenInternalCoHom</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X7A7FC1957A1905C6">1.8-18 InternalCoHomToTensorProductLeftAdjunctMorphism</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X85F970897F87BF2E">1.8-19 InternalCoHomToTensorProductLeftAdjunctMorphismWithGivenTensorProduct</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X8337DE6B7D61EAC2">1.8-20 MonoidalPreCoComposeMorphism</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X82A183B0807A0A01">1.8-21 MonoidalPreCoComposeMorphismWithGivenObjects</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X83FE0B8D853A76C1">1.8-22 MonoidalPostCoComposeMorphism</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X805DF38880B1E9E0">1.8-23 MonoidalPostCoComposeMorphismWithGivenObjects</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X817A1F7986256461">1.8-24 CoDualOnObjects</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X840CBC837926138E">1.8-25 CoDualOnMorphisms</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X79750BFD824D2AAF">1.8-26 CoDualOnMorphismsWithGivenCoDuals</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X7C65C05A788415E4">1.8-27 CoclosedEvaluationForCoDual</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X7CAB615C86D97CE8">1.8-28 CoclosedEvaluationForCoDualWithGivenTensorProduct</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X7841B6757A510799">1.8-29 MorphismFromCoBidual</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X7B2B002979E1CBF6">1.8-30 MorphismFromCoBidualWithGivenCoBidual</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X7D9E57197C820E0E">1.8-31 InternalCoHomTensorProductCompatibilityMorphism</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X81E9DC8D7FCD361F">1.8-32 InternalCoHomTensorProductCompatibilityMorphismWithGivenObjects</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X7B817BC97C1F1DF8">1.8-33 CoDualityTensorProductCompatibilityMorphism</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X82119ED686D3874C">1.8-34 CoDualityTensorProductCompatibilityMorphismWithGivenObjects</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X7A4971267B80B14F">1.8-35 MorphismFromInternalCoHomToTensorProduct</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X7B60FB927C37E125">1.8-36 MorphismFromInternalCoHomToTensorProductWithGivenObjects</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X86C7E40C85318EFA">1.8-37 IsomorphismFromCoDualObjectToInternalCoHomFromTensorUnit</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X84C86E29800BB8BC">1.8-38 IsomorphismFromInternalCoHomFromTensorUnitToCoDualObject</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X7AEE412D80799D09">1.8-39 UniversalPropertyOfCoDual</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X784CBFB984E66E7A">1.8-40 CoLambdaIntroduction</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X83520B098068CF62">1.8-41 CoLambdaElimination</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X83F11F6184DBD507">1.8-42 IsomorphismFromObjectToInternalCoHom</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X8233C995828387E9">1.8-43 IsomorphismFromObjectToInternalCoHomWithGivenInternalCoHom</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X875A349280E2095C">1.8-44 IsomorphismFromInternalCoHomToObject</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X858AD1F986BA5BB2">1.8-45 IsomorphismFromInternalCoHomToObjectWithGivenInternalCoHom</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap1_mj.html#X849F4CB58466EAEB">1.9 <span class="Heading">Symmetric Closed Monoidal Categories</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap1_mj.html#X86F60BCA79C63F20">1.10 <span class="Heading">Symmetric Coclosed Monoidal Categories</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap1_mj.html#X805AEF9784062A31">1.11 <span class="Heading">Rigid Symmetric Closed Monoidal Categories</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X782B629D7E7835C9">1.11-1 IsomorphismFromTensorProductWithDualObjectToInternalHom</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X84D0668C7CA0B63D">1.11-2 IsomorphismFromInternalHomToTensorProductWithDualObject</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X7D189B8280CECBA2">1.11-3 MorphismFromInternalHomToTensorProduct</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X7D10805D840AAC8D">1.11-4 MorphismFromInternalHomToTensorProductWithGivenObjects</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X85BA8C10817296F7">1.11-5 TensorProductInternalHomCompatibilityMorphismInverse</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X7CCE67B281EF45C1">1.11-6 TensorProductInternalHomCompatibilityMorphismInverseWithGivenObjects</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X83849B327C8074E9">1.11-7 CoevaluationForDual</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X7C65E6A97AAE0DE3">1.11-8 CoevaluationForDualWithGivenTensorProduct</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X85D0C9487A22AFFE">1.11-9 TraceMap</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X82F0DBD485D93793">1.11-10 RankMorphism</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X7E02E8197EA201EA">1.11-11 MorphismFromBidual</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X785CF0BB7BC0AC0D">1.11-12 MorphismFromBidualWithGivenBidual</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap1_mj.html#X79E86CAD853AB883">1.12 <span class="Heading">Rigid Symmetric Coclosed Monoidal Categories</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X82D4EF587F1C194C">1.12-1 IsomorphismFromInternalCoHomToTensorProductWithCoDualObject</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X7FE736BA834D228A">1.12-2 IsomorphismFromTensorProductWithCoDualObjectToInternalCoHom</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X7DCA54857A6B45DF">1.12-3 MorphismFromTensorProductToInternalCoHom</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X7888F9947DDC15B5">1.12-4 MorphismFromTensorProductToInternalCoHomWithGivenObjects</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X7E71AF0985C5AEC5">1.12-5 InternalCoHomTensorProductCompatibilityMorphismInverse</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X87BF8BE67AD5ABCF">1.12-6 InternalCoHomTensorProductCompatibilityMorphismInverseWithGivenObjects</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X8695F9D97A8C6C61">1.12-7 CoclosedCoevaluationForCoDual</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X78B2703A7B81E340">1.12-8 CoclosedCoevaluationForCoDualWithGivenTensorProduct</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X829BE5F97A656200">1.12-9 CoTraceMap</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X7C5CC8F97D95AD43">1.12-10 CoRankMorphism</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X85BDB94D85C67725">1.12-11 MorphismToCoBidual</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X800035BF867418D0">1.12-12 MorphismToCoBidualWithGivenCoBidual</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap1_mj.html#X7B40ED8B78D067A5">1.13 <span class="Heading">Convenience Methods</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X8055FF847AC2102A">1.13-1 InternalHom</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X7CB6A9497A971F59">1.13-2 InternalCoHom</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X82DE5FDA794914A0">1.13-3 LeftInternalHom</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X862E6A787C9E1F92">1.13-4 LeftInternalCoHom</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap1_mj.html#X85C8C80F785AEB5E">1.14 <span class="Heading">Add-methods</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X78DA7EC37A1E0CCC">1.14-1 AddLeftDistributivityExpanding</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X814BA200802D26E4">1.14-2 AddLeftDistributivityExpandingWithGivenObjects</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X8381C23A8264435B">1.14-3 AddLeftDistributivityFactoring</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X7F0A439478576973">1.14-4 AddLeftDistributivityFactoringWithGivenObjects</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X82E93F8E79CB7338">1.14-5 AddRightDistributivityExpanding</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X7A6178C17EFC817A">1.14-6 AddRightDistributivityExpandingWithGivenObjects</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X79B283777BC5F12C">1.14-7 AddRightDistributivityFactoring</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X842099557CF2036E">1.14-8 AddRightDistributivityFactoringWithGivenObjects</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X7F9C3FD38397D9D4">1.14-9 AddBraiding</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X8423AA1A862E9780">1.14-10 AddBraidingInverse</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X85C7E40583A5955F">1.14-11 AddBraidingInverseWithGivenTensorProducts</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X7A52CBE8801D6B28">1.14-12 AddBraidingWithGivenTensorProducts</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X7996A3A980BD5783">1.14-13 AddClosedMonoidalLeftCoevaluationMorphism</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X7FD9DC1D870FB3D2">1.14-14 AddClosedMonoidalLeftCoevaluationMorphismWithGivenRange</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X82E0E46E7B374DA9">1.14-15 AddClosedMonoidalLeftEvaluationMorphism</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X8742C27B7D30E543">1.14-16 AddClosedMonoidalLeftEvaluationMorphismWithGivenSource</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X870332B286200F2A">1.14-17 AddClosedMonoidalRightCoevaluationMorphism</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X7A06D2CC79874E18">1.14-18 AddClosedMonoidalRightCoevaluationMorphismWithGivenRange</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X82FD88E87B30334F">1.14-19 AddClosedMonoidalRightEvaluationMorphism</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X7ECDA3E47F267341">1.14-20 AddClosedMonoidalRightEvaluationMorphismWithGivenSource</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X800DFCF37CC10CCC">1.14-21 AddDualOnMorphisms</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X85DA12C581E673DE">1.14-22 AddDualOnMorphismsWithGivenDuals</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X7BB4DD5381EBF082">1.14-23 AddDualOnObjects</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X7FF9C96186646B7C">1.14-24 AddEvaluationForDual</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X7CB63A4E7E905CD4">1.14-25 AddEvaluationForDualWithGivenTensorProduct</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X79A5E1EF7F091948">1.14-26 AddInternalHomOnMorphisms</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X8070203E80388349">1.14-27 AddInternalHomOnMorphismsWithGivenInternalHoms</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X7B74F8FD8348D590">1.14-28 AddInternalHomOnObjects</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X87D40CEF8247DF33">1.14-29 AddInternalHomToTensorProductLeftAdjunctMorphism</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X82D3E7BD7DD67116">1.14-30 AddInternalHomToTensorProductLeftAdjunctMorphismWithGivenTensorProduct</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X7BBC91088476613F">1.14-31 AddInternalHomToTensorProductLeftAdjunctionIsomorphism</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X789D0ED680E7D176">1.14-32 AddInternalHomToTensorProductLeftAdjunctionIsomorphismWithGivenObjects</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X84B3BDBF8294E790">1.14-33 AddInternalHomToTensorProductRightAdjunctMorphism</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X807104E57D50EEB9">1.14-34 AddInternalHomToTensorProductRightAdjunctMorphismWithGivenTensorProduct</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X8634234A855F18D5">1.14-35 AddInternalHomToTensorProductRightAdjunctionIsomorphism</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X7A3FED8E84CD4A02">1.14-36 AddInternalHomToTensorProductRightAdjunctionIsomorphismWithGivenObjects</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X7983CB57783E6813">1.14-37 AddIsomorphismFromDualObjectToInternalHomIntoTensorUnit</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X84E4949886CB84C9">1.14-38 AddIsomorphismFromInternalHomIntoTensorUnitToDualObject</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X7FA9FE2B85CE866D">1.14-39 AddIsomorphismFromInternalHomToObject</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X7D6421147BF6651B">1.14-40 AddIsomorphismFromInternalHomToObjectWithGivenInternalHom</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X82538F687DE872DD">1.14-41 AddIsomorphismFromObjectToInternalHom</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X78ED7FCC83D091AB">1.14-42 AddIsomorphismFromObjectToInternalHomWithGivenInternalHom</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X8406FBCD7EE968FF">1.14-43 AddLambdaElimination</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X84944B6283F2802A">1.14-44 AddLambdaIntroduction</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X843DC64486CB2ED1">1.14-45 AddMonoidalPostComposeMorphism</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X7FC3725A7CF804F9">1.14-46 AddMonoidalPostComposeMorphismWithGivenObjects</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X7B0C198782C01CBA">1.14-47 AddMonoidalPreComposeMorphism</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X7A22CD657A2A338C">1.14-48 AddMonoidalPreComposeMorphismWithGivenObjects</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X7CF664D87D8D5CA1">1.14-49 AddMorphismFromTensorProductToInternalHom</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X86D3AF4A80FA738D">1.14-50 AddMorphismFromTensorProductToInternalHomWithGivenObjects</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X864B98B08041EE6C">1.14-51 AddMorphismToBidual</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X84C05B087EFCE599">1.14-52 AddMorphismToBidualWithGivenBidual</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X7A020DBF860317F5">1.14-53 AddTensorProductDualityCompatibilityMorphism</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X78D1B6AF8654A950">1.14-54 AddTensorProductDualityCompatibilityMorphismWithGivenObjects</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X83A6EF0B7ED71EED">1.14-55 AddTensorProductInternalHomCompatibilityMorphism</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X871C6B55843A0AF3">1.14-56 AddTensorProductInternalHomCompatibilityMorphismWithGivenObjects</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X817EAD878544BF68">1.14-57 AddTensorProductToInternalHomLeftAdjunctMorphism</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X7B2FEF0985C2FACE">1.14-58 AddTensorProductToInternalHomLeftAdjunctMorphismWithGivenInternalHom</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X85A8D11983750164">1.14-59 AddTensorProductToInternalHomLeftAdjunctionIsomorphism</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X7AE873FD87E4B12D">1.14-60 AddTensorProductToInternalHomLeftAdjunctionIsomorphismWithGivenObjects</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X7AF5FBEE859787CB">1.14-61 AddTensorProductToInternalHomRightAdjunctMorphism</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X86B3F2BF813AFBDA">1.14-62 AddTensorProductToInternalHomRightAdjunctMorphismWithGivenInternalHom</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X79777C5F825C788E">1.14-63 AddTensorProductToInternalHomRightAdjunctionIsomorphism</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X86FD5F8A83CE2A59">1.14-64 AddTensorProductToInternalHomRightAdjunctionIsomorphismWithGivenObjects</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X7FCEDB3B7DF69A1E">1.14-65 AddUniversalPropertyOfDual</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X7D23088385F021AA">1.14-66 AddCoDualOnMorphisms</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X7C2568FD82259231">1.14-67 AddCoDualOnMorphismsWithGivenCoDuals</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X7C1CDE7F855ACB74">1.14-68 AddCoDualOnObjects</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X7D9B355D8261CB17">1.14-69 AddCoDualityTensorProductCompatibilityMorphism</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X7CE8010980DBA597">1.14-70 AddCoDualityTensorProductCompatibilityMorphismWithGivenObjects</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X7E7962CC7E809023">1.14-71 AddCoLambdaElimination</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X7C3868657D8A0400">1.14-72 AddCoLambdaIntroduction</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X839204FD79F2968D">1.14-73 AddCoclosedEvaluationForCoDual</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X803976DB7C8F307C">1.14-74 AddCoclosedEvaluationForCoDualWithGivenTensorProduct</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X829CAF887FAC5BF0">1.14-75 AddCoclosedMonoidalLeftCoevaluationMorphism</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X7E7EBA4381366E7A">1.14-76 AddCoclosedMonoidalLeftCoevaluationMorphismWithGivenSource</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X81EC6B3E80876BC8">1.14-77 AddCoclosedMonoidalLeftEvaluationMorphism</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X839DE15C87358F99">1.14-78 AddCoclosedMonoidalLeftEvaluationMorphismWithGivenRange</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X78A8CB687ADC4653">1.14-79 AddCoclosedMonoidalRightCoevaluationMorphism</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X783C75CD79B2E8C9">1.14-80 AddCoclosedMonoidalRightCoevaluationMorphismWithGivenSource</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X85F82B2F84E55DD2">1.14-81 AddCoclosedMonoidalRightEvaluationMorphism</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X86DC2EF07B421CE0">1.14-82 AddCoclosedMonoidalRightEvaluationMorphismWithGivenRange</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X7D63C4A57A467DAA">1.14-83 AddInternalCoHomOnMorphisms</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X87A67CDD86E0CE05">1.14-84 AddInternalCoHomOnMorphismsWithGivenInternalCoHoms</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X828648F183570AE3">1.14-85 AddInternalCoHomOnObjects</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X78278696834ACA53">1.14-86 AddInternalCoHomTensorProductCompatibilityMorphism</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X82EA4C857C7BCC60">1.14-87 AddInternalCoHomTensorProductCompatibilityMorphismWithGivenObjects</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X7C7256EE79374540">1.14-88 AddInternalCoHomToTensorProductLeftAdjunctMorphism</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X84EBD01D7843FAAD">1.14-89 AddInternalCoHomToTensorProductLeftAdjunctMorphismWithGivenTensorProduct</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X858ED26087369677">1.14-90 AddInternalCoHomToTensorProductRightAdjunctMorphism</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X7E97DAC0782AACEC">1.14-91 AddInternalCoHomToTensorProductRightAdjunctMorphismWithGivenTensorProduct</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X7A2B14497B4924A7">1.14-92 AddIsomorphismFromCoDualObjectToInternalCoHomFromTensorUnit</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X8143326C7BE5E321">1.14-93 AddIsomorphismFromInternalCoHomFromTensorUnitToCoDualObject</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X7C9F35A779395B26">1.14-94 AddIsomorphismFromInternalCoHomToObject</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X7D0E88AC7C0304D3">1.14-95 AddIsomorphismFromInternalCoHomToObjectWithGivenInternalCoHom</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X7E18F31C86620139">1.14-96 AddIsomorphismFromObjectToInternalCoHom</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X8304276383585ECC">1.14-97 AddIsomorphismFromObjectToInternalCoHomWithGivenInternalCoHom</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X7DE479077F155716">1.14-98 AddMonoidalPostCoComposeMorphism</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X79858B577BD4F1E1">1.14-99 AddMonoidalPostCoComposeMorphismWithGivenObjects</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X7D1662EA819EA1A2">1.14-100 AddMonoidalPreCoComposeMorphism</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X81DFED7386A953E0">1.14-101 AddMonoidalPreCoComposeMorphismWithGivenObjects</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X7D1615257BAA311F">1.14-102 AddMorphismFromCoBidual</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X84D1F74D8118B2A0">1.14-103 AddMorphismFromCoBidualWithGivenCoBidual</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X7C3C01F1834A4B9B">1.14-104 AddMorphismFromInternalCoHomToTensorProduct</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X87AB2C4D84F19F37">1.14-105 AddMorphismFromInternalCoHomToTensorProductWithGivenObjects</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X7C0E0CF87FEFEB0D">1.14-106 AddTensorProductToInternalCoHomLeftAdjunctMorphism</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X7A838651781657DA">1.14-107 AddTensorProductToInternalCoHomLeftAdjunctMorphismWithGivenInternalCoHom</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X8123546E81EE383A">1.14-108 AddTensorProductToInternalCoHomRightAdjunctMorphism</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X8698898584E99695">1.14-109 AddTensorProductToInternalCoHomRightAdjunctMorphismWithGivenInternalCoHom</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X8519ECF17EC07916">1.14-110 AddUniversalPropertyOfCoDual</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X7F868E597F1C1AD7">1.14-111 AddIsomorphismFromLeftDualObjectToLeftInternalHomIntoTensorUnit</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X810B50C27901021C">1.14-112 AddIsomorphismFromLeftInternalHomIntoTensorUnitToLeftDualObject</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X87415032853E8340">1.14-113 AddIsomorphismFromLeftInternalHomToObject</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X83DAADEE8139DACA">1.14-114 AddIsomorphismFromLeftInternalHomToObjectWithGivenLeftInternalHom</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X7FA337D07BCD9D5B">1.14-115 AddIsomorphismFromObjectToLeftInternalHom</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X7EA7B1777FCAC4D1">1.14-116 AddIsomorphismFromObjectToLeftInternalHomWithGivenLeftInternalHom</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X7D1E4F66851991AB">1.14-117 AddLeftClosedMonoidalCoevaluationMorphism</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X7EF0D5B782AB75FA">1.14-118 AddLeftClosedMonoidalCoevaluationMorphismWithGivenRange</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X7FD8142C7CBF7FB1">1.14-119 AddLeftClosedMonoidalEvaluationForLeftDual</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X7A89130185CCF31D">1.14-120 AddLeftClosedMonoidalEvaluationForLeftDualWithGivenTensorProduct</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X7D75530C7E938B81">1.14-121 AddLeftClosedMonoidalEvaluationMorphism</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X8535FD8E7894236B">1.14-122 AddLeftClosedMonoidalEvaluationMorphismWithGivenSource</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X869B3FD87FC09E21">1.14-123 AddLeftClosedMonoidalLambdaElimination</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X7CC085AB7DBC581A">1.14-124 AddLeftClosedMonoidalLambdaIntroduction</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X7BE971CB7E989544">1.14-125 AddLeftClosedMonoidalPostComposeMorphism</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X7B6DBF17796A971A">1.14-126 AddLeftClosedMonoidalPostComposeMorphismWithGivenObjects</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X799401467830D201">1.14-127 AddLeftClosedMonoidalPreComposeMorphism</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X7915D4437C9257A5">1.14-128 AddLeftClosedMonoidalPreComposeMorphismWithGivenObjects</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X79D85C017917C1D4">1.14-129 AddLeftDualOnMorphisms</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X816C625A82BCDDE3">1.14-130 AddLeftDualOnMorphismsWithGivenLeftDuals</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X7CB93B6E8555359D">1.14-131 AddLeftDualOnObjects</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X79B3B2F586530EFF">1.14-132 AddLeftInternalHomOnMorphisms</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X7DDC722B78B659A2">1.14-133 AddLeftInternalHomOnMorphismsWithGivenLeftInternalHoms</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X7FCF9DEA7F6B73F9">1.14-134 AddLeftInternalHomOnObjects</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X7F5EAC1E8104166E">1.14-135 AddLeftInternalHomToTensorProductAdjunctMorphism</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X7C437ED77E95B84B">1.14-136 AddLeftInternalHomToTensorProductAdjunctMorphismWithGivenTensorProduct</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X7CA082FF7B01D8FB">1.14-137 AddMorphismFromTensorProductToLeftInternalHom</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X8028DDA77A2AF1D0">1.14-138 AddMorphismFromTensorProductToLeftInternalHomWithGivenObjects</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X80EC71197C2C11D1">1.14-139 AddMorphismToLeftBidual</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X869265BC78942371">1.14-140 AddMorphismToLeftBidualWithGivenLeftBidual</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X7953DC1F805D8C1A">1.14-141 AddTensorProductLeftDualityCompatibilityMorphism</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X836F9F347AB09804">1.14-142 AddTensorProductLeftDualityCompatibilityMorphismWithGivenObjects</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X7A76B21F7DD29DD4">1.14-143 AddTensorProductLeftInternalHomCompatibilityMorphism</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X78BE6DC079995BE8">1.14-144 AddTensorProductLeftInternalHomCompatibilityMorphismWithGivenObjects</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X7A7B64B17E0CB68F">1.14-145 AddTensorProductToLeftInternalHomAdjunctMorphism</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X852D330679DA479F">1.14-146 AddTensorProductToLeftInternalHomAdjunctMorphismWithGivenLeftInternalHom</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X814F44BD7ED47029">1.14-147 AddUniversalPropertyOfLeftDual</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X7DF0FCAF7EF18F8F">1.14-148 AddIsomorphismFromLeftCoDualObjectToLeftInternalCoHomFromTensorUnit</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X85BAFAF48192ADFD">1.14-149 AddIsomorphismFromLeftInternalCoHomFromTensorUnitToLeftCoDualObject</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X7F13CA18790E4170">1.14-150 AddIsomorphismFromLeftInternalCoHomToObject</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X8695E6B0854AC59F">1.14-151 AddIsomorphismFromLeftInternalCoHomToObjectWithGivenLeftInternalCoHom</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X78B4B7FE82DC4548">1.14-152 AddIsomorphismFromObjectToLeftInternalCoHom</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X8569A8107E98C1A7">1.14-153 AddIsomorphismFromObjectToLeftInternalCoHomWithGivenLeftInternalCoHom</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X834FFF597E5AD96E">1.14-154 AddLeftCoDualOnMorphisms</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X7EFAFD687AE8B744">1.14-155 AddLeftCoDualOnMorphismsWithGivenLeftCoDuals</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X85C97E8D7982AA91">1.14-156 AddLeftCoDualOnObjects</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X859C18CA80B931C8">1.14-157 AddLeftCoDualityTensorProductCompatibilityMorphism</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X7D8D0BE17F8837FB">1.14-158 AddLeftCoDualityTensorProductCompatibilityMorphismWithGivenObjects</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X788B63417F1F60F9">1.14-159 AddLeftCoclosedMonoidalCoevaluationMorphism</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X82B5F67881855573">1.14-160 AddLeftCoclosedMonoidalCoevaluationMorphismWithGivenSource</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X7825101881671E7F">1.14-161 AddLeftCoclosedMonoidalEvaluationForLeftCoDual</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X846EE69A797450F9">1.14-162 AddLeftCoclosedMonoidalEvaluationForLeftCoDualWithGivenTensorProduct</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X826AE548803450C1">1.14-163 AddLeftCoclosedMonoidalEvaluationMorphism</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X8082909C8786B490">1.14-164 AddLeftCoclosedMonoidalEvaluationMorphismWithGivenRange</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X84635F4580F63BD0">1.14-165 AddLeftCoclosedMonoidalLambdaElimination</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X83DF33EF7A19E291">1.14-166 AddLeftCoclosedMonoidalLambdaIntroduction</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X7954E35E784BCB8B">1.14-167 AddLeftCoclosedMonoidalPostCoComposeMorphism</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X7FE2EC92781C752E">1.14-168 AddLeftCoclosedMonoidalPostCoComposeMorphismWithGivenObjects</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X7DC17F0884BAD720">1.14-169 AddLeftCoclosedMonoidalPreCoComposeMorphism</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X86DB94FE8301038C">1.14-170 AddLeftCoclosedMonoidalPreCoComposeMorphismWithGivenObjects</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X829E5F7E783539C3">1.14-171 AddLeftInternalCoHomOnMorphisms</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X78538C8E78635666">1.14-172 AddLeftInternalCoHomOnMorphismsWithGivenLeftInternalCoHoms</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X82901BEB81D0A428">1.14-173 AddLeftInternalCoHomOnObjects</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X814CCC847F369708">1.14-174 AddLeftInternalCoHomTensorProductCompatibilityMorphism</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X7C3A3DEB7BA72741">1.14-175 AddLeftInternalCoHomTensorProductCompatibilityMorphismWithGivenObjects</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X80C1A62B812B4565">1.14-176 AddLeftInternalCoHomToTensorProductAdjunctMorphism</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X83DDE598805FFA88">1.14-177 AddLeftInternalCoHomToTensorProductAdjunctMorphismWithGivenTensorProduct</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X8104426A8327EF5E">1.14-178 AddMorphismFromLeftCoBidual</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X849C36147F09B8A0">1.14-179 AddMorphismFromLeftCoBidualWithGivenLeftCoBidual</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X86168FF07A697A15">1.14-180 AddMorphismFromLeftInternalCoHomToTensorProduct</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X830428C27A6BC07B">1.14-181 AddMorphismFromLeftInternalCoHomToTensorProductWithGivenObjects</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X7AA7EDE881098270">1.14-182 AddTensorProductToLeftInternalCoHomAdjunctMorphism</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X7CDB23877B3C8292">1.14-183 AddTensorProductToLeftInternalCoHomAdjunctMorphismWithGivenLeftInternalCoHom</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X864F1EF47CAD2BE8">1.14-184 AddUniversalPropertyOfLeftCoDual</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X79CDE3B87D14EF9A">1.14-185 AddAssociatorLeftToRight</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X850B5B357F592FAE">1.14-186 AddAssociatorLeftToRightWithGivenTensorProducts</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X8045C4FC7B4912B4">1.14-187 AddAssociatorRightToLeft</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X7CF105217904D280">1.14-188 AddAssociatorRightToLeftWithGivenTensorProducts</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X7986052487D3CDBB">1.14-189 AddLeftUnitor</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X793D2C657F19096D">1.14-190 AddLeftUnitorInverse</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X87CB487587ED3EC5">1.14-191 AddLeftUnitorInverseWithGivenTensorProduct</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X8614FD9C78812C98">1.14-192 AddLeftUnitorWithGivenTensorProduct</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X8568708E7C5D8D28">1.14-193 AddRightUnitor</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X7B2AAD6286833F68">1.14-194 AddRightUnitorInverse</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X8682171F86C6793E">1.14-195 AddRightUnitorInverseWithGivenTensorProduct</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X7DB39DC9794F3FCA">1.14-196 AddRightUnitorWithGivenTensorProduct</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X83DEDFA3803035BC">1.14-197 AddTensorProductOnMorphisms</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X879659397D11AD9F">1.14-198 AddTensorProductOnMorphismsWithGivenTensorProducts</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X7CED389B792142F5">1.14-199 AddTensorProductOnObjects</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X7BC4B940858F903B">1.14-200 AddTensorUnit</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X8586506084917A30">1.14-201 AddCoevaluationForDual</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X831DC3947941E8BA">1.14-202 AddCoevaluationForDualWithGivenTensorProduct</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X817D08298188F2BC">1.14-203 AddIsomorphismFromInternalHomToTensorProductWithDualObject</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X840BC8AB7DFFD68E">1.14-204 AddIsomorphismFromTensorProductWithDualObjectToInternalHom</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X7D2FDE687E0CB66A">1.14-205 AddMorphismFromBidual</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X877C409E865C2BB3">1.14-206 AddMorphismFromBidualWithGivenBidual</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X82FDEF277A71FABF">1.14-207 AddMorphismFromInternalHomToTensorProduct</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X7A4A4C668706D593">1.14-208 AddMorphismFromInternalHomToTensorProductWithGivenObjects</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X849B17E08679450A">1.14-209 AddRankMorphism</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X798FA16B818559DB">1.14-210 AddTensorProductInternalHomCompatibilityMorphismInverse</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X7D9EA11980170B0C">1.14-211 AddTensorProductInternalHomCompatibilityMorphismInverseWithGivenObjects</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X8446431880FFF111">1.14-212 AddTraceMap</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X7A3D91F87F170676">1.14-213 AddCoRankMorphism</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X7AA8757780D75E1F">1.14-214 AddCoTraceMap</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X82087A8C7E5EBAA8">1.14-215 AddCoclosedCoevaluationForCoDual</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X813CF5C07BAFD19C">1.14-216 AddCoclosedCoevaluationForCoDualWithGivenTensorProduct</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X7CA7972381FC507F">1.14-217 AddInternalCoHomTensorProductCompatibilityMorphismInverse</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X82F222EC80852CF1">1.14-218 AddInternalCoHomTensorProductCompatibilityMorphismInverseWithGivenObjects</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X84E2048279A169A6">1.14-219 AddIsomorphismFromInternalCoHomToTensorProductWithCoDualObject</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X7DFE79B48125A1E0">1.14-220 AddIsomorphismFromTensorProductWithCoDualObjectToInternalCoHom</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X7F27D275849C84B2">1.14-221 AddMorphismFromTensorProductToInternalCoHom</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X7B5706318327501E">1.14-222 AddMorphismFromTensorProductToInternalCoHomWithGivenObjects</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X7EBD7033871B0F3D">1.14-223 AddMorphismToCoBidual</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1_mj.html#X867E02E07CEC5B34">1.14-224 AddMorphismToCoBidualWithGivenCoBidual</a></span>
</div></div>
</div>
<h3>1 <span class="Heading">Monoidal Categories</span></h3>
<p><a id="X808DC49C7ED99B52" name="X808DC49C7ED99B52"></a></p>
<h4>1.1 <span class="Heading">Monoidal Categories</span></h4>
<p>A <span class="SimpleMath">\(6\)</span>-tuple <span class="SimpleMath">\(( \mathbf{C}, \otimes, 1, \alpha, \lambda, \rho )\)</span> consisting of</p>
<ul>
<li><p>a category <span class="SimpleMath">\(\mathbf{C}\)</span>,</p>
</li>
<li><p>a functor <span class="SimpleMath">\(\otimes: \mathbf{C} \times \mathbf{C} \rightarrow \mathbf{C}\)</span> compatible with the congruence of morphisms,</p>
</li>
<li><p>an object <span class="SimpleMath">\(1 \in \mathbf{C}\)</span>,</p>
</li>
<li><p>a natural isomorphism <span class="SimpleMath">\(\alpha_{a,b,c}: a \otimes (b \otimes c) \cong (a \otimes b) \otimes c\)</span>,</p>
</li>
<li><p>a natural isomorphism <span class="SimpleMath">\(\lambda_{a}: 1 \otimes a \cong a\)</span>,</p>
</li>
<li><p>a natural isomorphism <span class="SimpleMath">\(\rho_{a}: a \otimes 1 \cong a\)</span>,</p>
</li>
</ul>
<p>is called a <em>monoidal category</em>, if</p>
<ul>
<li><p>for all objects <span class="SimpleMath">\(a,b,c,d\)</span>, the pentagon identity holds:</p>
</li>
</ul>
<p><span class="SimpleMath">\((\alpha_{a,b,c} \otimes \mathrm{id}_d) \circ \alpha_{a,b \otimes c, d} \circ ( \mathrm{id}_a \otimes \alpha_{b,c,d} ) \sim \alpha_{a \otimes b, c, d} \circ \alpha_{a,b,c \otimes d}\)</span>,</p>
<ul>
<li><p>for all objects <span class="SimpleMath">\(a,c\)</span>, the triangle identity holds:</p>
</li>
</ul>
<p><span class="SimpleMath">\(( \rho_a \otimes \mathrm{id}_c ) \circ \alpha_{a,1,c} \sim \mathrm{id}_a \otimes \lambda_c\)</span>.</p>
<p>The corresponding GAP property is given by <code class="code">IsMonoidalCategory</code>.</p>
<p><a id="X84997A1E8188D6BE" name="X84997A1E8188D6BE"></a></p>
<h5>1.1-1 TensorProductOnMorphisms</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ TensorProductOnMorphisms</code>( <var class="Arg">alpha</var>, <var class="Arg">beta</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: a morphism in <span class="SimpleMath">\(\mathrm{Hom}(a \otimes b, a' \otimes b')\)</span></p>
<p>The arguments are two morphisms <span class="SimpleMath">\(\alpha: a \rightarrow a', \beta: b \rightarrow b'\)</span>. The output is the tensor product <span class="SimpleMath">\(\alpha \otimes \beta\)</span>.</p>
<p><a id="X85008BF07C2C0386" name="X85008BF07C2C0386"></a></p>
<h5>1.1-2 TensorProductOnMorphismsWithGivenTensorProducts</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ TensorProductOnMorphismsWithGivenTensorProducts</code>( <var class="Arg">s</var>, <var class="Arg">alpha</var>, <var class="Arg">beta</var>, <var class="Arg">r</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: a morphism in <span class="SimpleMath">\(\mathrm{Hom}(a \otimes b, a' \otimes b')\)</span></p>
<p>The arguments are an object <span class="SimpleMath">\(s = a \otimes b\)</span>, two morphisms <span class="SimpleMath">\(\alpha: a \rightarrow a', \beta: b \rightarrow b'\)</span>, and an object <span class="SimpleMath">\(r = a' \otimes b'\)</span>. The output is the tensor product <span class="SimpleMath">\(\alpha \otimes \beta\)</span>.</p>
<p><a id="X84C2FAEA7B9678AA" name="X84C2FAEA7B9678AA"></a></p>
<h5>1.1-3 AssociatorRightToLeft</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ AssociatorRightToLeft</code>( <var class="Arg">a</var>, <var class="Arg">b</var>, <var class="Arg">c</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: a morphism in <span class="SimpleMath">\(\mathrm{Hom}( a \otimes (b \otimes c), (a \otimes b) \otimes c )\)</span>.</p>
<p>The arguments are three objects <span class="SimpleMath">\(a,b,c\)</span>. The output is the associator <span class="SimpleMath">\(\alpha_{a,(b,c)}: a \otimes (b \otimes c) \rightarrow (a \otimes b) \otimes c\)</span>.</p>
<p><a id="X85645C4D86060ED2" name="X85645C4D86060ED2"></a></p>
<h5>1.1-4 AssociatorRightToLeftWithGivenTensorProducts</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ AssociatorRightToLeftWithGivenTensorProducts</code>( <var class="Arg">s</var>, <var class="Arg">a</var>, <var class="Arg">b</var>, <var class="Arg">c</var>, <var class="Arg">r</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: a morphism in <span class="SimpleMath">\(\mathrm{Hom}( a \otimes (b \otimes c), (a \otimes b) \otimes c )\)</span>.</p>
<p>The arguments are an object <span class="SimpleMath">\(s = a \otimes (b \otimes c)\)</span>, three objects <span class="SimpleMath">\(a,b,c\)</span>, and an object <span class="SimpleMath">\(r = (a \otimes b) \otimes c\)</span>. The output is the associator <span class="SimpleMath">\(\alpha_{a,(b,c)}: a \otimes (b \otimes c) \rightarrow (a \otimes b) \otimes c\)</span>.</p>
<p><a id="X7C5A064C7E1995F1" name="X7C5A064C7E1995F1"></a></p>
<h5>1.1-5 AssociatorLeftToRight</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ AssociatorLeftToRight</code>( <var class="Arg">a</var>, <var class="Arg">b</var>, <var class="Arg">c</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: a morphism in <span class="SimpleMath">\(\mathrm{Hom}( (a \otimes b) \otimes c \rightarrow a \otimes (b \otimes c) )\)</span>.</p>
<p>The arguments are three objects <span class="SimpleMath">\(a,b,c\)</span>. The output is the associator <span class="SimpleMath">\(\alpha_{(a,b),c}: (a \otimes b) \otimes c \rightarrow a \otimes (b \otimes c)\)</span>.</p>
<p><a id="X7C59D73F8389E389" name="X7C59D73F8389E389"></a></p>
<h5>1.1-6 AssociatorLeftToRightWithGivenTensorProducts</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ AssociatorLeftToRightWithGivenTensorProducts</code>( <var class="Arg">s</var>, <var class="Arg">a</var>, <var class="Arg">b</var>, <var class="Arg">c</var>, <var class="Arg">r</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: a morphism in <span class="SimpleMath">\(\mathrm{Hom}( (a \otimes b) \otimes c \rightarrow a \otimes (b \otimes c) )\)</span>.</p>
<p>The arguments are an object <span class="SimpleMath">\(s = (a \otimes b) \otimes c\)</span>, three objects <span class="SimpleMath">\(a,b,c\)</span>, and an object <span class="SimpleMath">\(r = a \otimes (b \otimes c)\)</span>. The output is the associator <span class="SimpleMath">\(\alpha_{(a,b),c}: (a \otimes b) \otimes c \rightarrow a \otimes (b \otimes c)\)</span>.</p>
<p><a id="X7EBCAFDD81BDB655" name="X7EBCAFDD81BDB655"></a></p>
<h5>1.1-7 LeftUnitor</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ LeftUnitor</code>( <var class="Arg">a</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: a morphism in <span class="SimpleMath">\(\mathrm{Hom}(1 \otimes a, a)\)</span></p>
<p>The argument is an object <span class="SimpleMath">\(a\)</span>. The output is the left unitor <span class="SimpleMath">\(\lambda_a: 1 \otimes a \rightarrow a\)</span>.</p>
<p><a id="X7ED284DF7BA21774" name="X7ED284DF7BA21774"></a></p>
<h5>1.1-8 LeftUnitorWithGivenTensorProduct</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ LeftUnitorWithGivenTensorProduct</code>( <var class="Arg">a</var>, <var class="Arg">s</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: a morphism in <span class="SimpleMath">\(\mathrm{Hom}(1 \otimes a, a)\)</span></p>
<p>The arguments are an object <span class="SimpleMath">\(a\)</span> and an object <span class="SimpleMath">\(s = 1 \otimes a\)</span>. The output is the left unitor <span class="SimpleMath">\(\lambda_a: 1 \otimes a \rightarrow a\)</span>.</p>
<p><a id="X79A274078147A9F7" name="X79A274078147A9F7"></a></p>
<h5>1.1-9 LeftUnitorInverse</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ LeftUnitorInverse</code>( <var class="Arg">a</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: a morphism in <span class="SimpleMath">\(\mathrm{Hom}(a, 1 \otimes a)\)</span></p>
<p>The argument is an object <span class="SimpleMath">\(a\)</span>. The output is the inverse of the left unitor <span class="SimpleMath">\(\lambda_a^{-1}: a \rightarrow 1 \otimes a\)</span>.</p>
<p><a id="X849609447CAE996B" name="X849609447CAE996B"></a></p>
<h5>1.1-10 LeftUnitorInverseWithGivenTensorProduct</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ LeftUnitorInverseWithGivenTensorProduct</code>( <var class="Arg">a</var>, <var class="Arg">r</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: a morphism in <span class="SimpleMath">\(\mathrm{Hom}(a, 1 \otimes a)\)</span></p>
<p>The argument is an object <span class="SimpleMath">\(a\)</span> and an object <span class="SimpleMath">\(r = 1 \otimes a\)</span>. The output is the inverse of the left unitor <span class="SimpleMath">\(\lambda_a^{-1}: a \rightarrow 1 \otimes a\)</span>.</p>
<p><a id="X82F758CB7A23468A" name="X82F758CB7A23468A"></a></p>
<h5>1.1-11 RightUnitor</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ RightUnitor</code>( <var class="Arg">a</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: a morphism in <span class="SimpleMath">\(\mathrm{Hom}(a \otimes 1, a)\)</span></p>
<p>The argument is an object <span class="SimpleMath">\(a\)</span>. The output is the right unitor <span class="SimpleMath">\(\rho_a: a \otimes 1 \rightarrow a\)</span>.</p>
<p><a id="X84C8F97F7BF2282D" name="X84C8F97F7BF2282D"></a></p>
<h5>1.1-12 RightUnitorWithGivenTensorProduct</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ RightUnitorWithGivenTensorProduct</code>( <var class="Arg">a</var>, <var class="Arg">s</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: a morphism in <span class="SimpleMath">\(\mathrm{Hom}(a \otimes 1, a)\)</span></p>
<p>The arguments are an object <span class="SimpleMath">\(a\)</span> and an object <span class="SimpleMath">\(s = a \otimes 1\)</span>. The output is the right unitor <span class="SimpleMath">\(\rho_a: a \otimes 1 \rightarrow a\)</span>.</p>
<p><a id="X82DC825480E87301" name="X82DC825480E87301"></a></p>
<h5>1.1-13 RightUnitorInverse</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ RightUnitorInverse</code>( <var class="Arg">a</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: a morphism in <span class="SimpleMath">\(\mathrm{Hom}(a, a \otimes 1)\)</span></p>
<p>The argument is an object <span class="SimpleMath">\(a\)</span>. The output is the inverse of the right unitor <span class="SimpleMath">\(\rho_a^{-1}: a \rightarrow a \otimes 1\)</span>.</p>
<p><a id="X85A8A87C87B33008" name="X85A8A87C87B33008"></a></p>
<h5>1.1-14 RightUnitorInverseWithGivenTensorProduct</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ RightUnitorInverseWithGivenTensorProduct</code>( <var class="Arg">a</var>, <var class="Arg">r</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: a morphism in <span class="SimpleMath">\(\mathrm{Hom}(a, a \otimes 1)\)</span></p>
<p>The arguments are an object <span class="SimpleMath">\(a\)</span> and an object <span class="SimpleMath">\(r = a \otimes 1\)</span>. The output is the inverse of the right unitor <span class="SimpleMath">\(\rho_a^{-1}: a \rightarrow a \otimes 1\)</span>.</p>
<p><a id="X797034F17AA47EE0" name="X797034F17AA47EE0"></a></p>
<h5>1.1-15 TensorProductOnObjects</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ TensorProductOnObjects</code>( <var class="Arg">a</var>, <var class="Arg">b</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: an object</p>
<p>The arguments are two objects <span class="SimpleMath">\(a, b\)</span>. The output is the tensor product <span class="SimpleMath">\(a \otimes b\)</span>.</p>
<p><a id="X7BF7FFDF8789474A" name="X7BF7FFDF8789474A"></a></p>
<h5>1.1-16 TensorUnit</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ TensorUnit</code>( <var class="Arg">C</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: an object</p>
<p>The argument is a category <span class="SimpleMath">\(\mathbf{C}\)</span>. The output is the tensor unit <span class="SimpleMath">\(1\)</span> of <span class="SimpleMath">\(\mathbf{C}\)</span>.</p>
<p><a id="X7FB8551A815BEC53" name="X7FB8551A815BEC53"></a></p>
<h4>1.2 <span class="Heading">Additive Monoidal Categories</span></h4>
<p><a id="X7ACF268E7D82B0C9" name="X7ACF268E7D82B0C9"></a></p>
<h5>1.2-1 LeftDistributivityExpanding</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ LeftDistributivityExpanding</code>( <var class="Arg">a</var>, <var class="Arg">L</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: a morphism in <span class="SimpleMath">\(\mathrm{Hom}( a \otimes (b_1 \oplus \dots \oplus b_n), (a \otimes b_1) \oplus \dots \oplus (a \otimes b_n) )\)</span></p>
<p>The arguments are an object <span class="SimpleMath">\(a\)</span> and a list of objects <span class="SimpleMath">\(L = (b_1, \dots, b_n)\)</span>. The output is the left distributivity morphism <span class="SimpleMath">\(a \otimes (b_1 \oplus \dots \oplus b_n) \rightarrow (a \otimes b_1) \oplus \dots \oplus (a \otimes b_n)\)</span>.</p>
<p><a id="X7CA18968862D2285" name="X7CA18968862D2285"></a></p>
<h5>1.2-2 LeftDistributivityExpandingWithGivenObjects</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ LeftDistributivityExpandingWithGivenObjects</code>( <var class="Arg">s</var>, <var class="Arg">a</var>, <var class="Arg">L</var>, <var class="Arg">r</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: a morphism in <span class="SimpleMath">\(\mathrm{Hom}( s, r )\)</span></p>
<p>The arguments are an object <span class="SimpleMath">\(s = a \otimes (b_1 \oplus \dots \oplus b_n)\)</span>, an object <span class="SimpleMath">\(a\)</span>, a list of objects <span class="SimpleMath">\(L = (b_1, \dots, b_n)\)</span>, and an object <span class="SimpleMath">\(r = (a \otimes b_1) \oplus \dots \oplus (a \otimes b_n)\)</span>. The output is the left distributivity morphism <span class="SimpleMath">\(s \rightarrow r\)</span>.</p>
<p><a id="X780322FE7E5A626D" name="X780322FE7E5A626D"></a></p>
<h5>1.2-3 LeftDistributivityFactoring</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ LeftDistributivityFactoring</code>( <var class="Arg">a</var>, <var class="Arg">L</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: a morphism in <span class="SimpleMath">\(\mathrm{Hom}( (a \otimes b_1) \oplus \dots \oplus (a \otimes b_n), a \otimes (b_1 \oplus \dots \oplus b_n) )\)</span></p>
<p>The arguments are an object <span class="SimpleMath">\(a\)</span> and a list of objects <span class="SimpleMath">\(L = (b_1, \dots, b_n)\)</span>. The output is the left distributivity morphism <span class="SimpleMath">\((a \otimes b_1) \oplus \dots \oplus (a \otimes b_n) \rightarrow a \otimes (b_1 \oplus \dots \oplus b_n)\)</span>.</p>
<p><a id="X784D8CD685F5F021" name="X784D8CD685F5F021"></a></p>
<h5>1.2-4 LeftDistributivityFactoringWithGivenObjects</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ LeftDistributivityFactoringWithGivenObjects</code>( <var class="Arg">s</var>, <var class="Arg">a</var>, <var class="Arg">L</var>, <var class="Arg">r</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: a morphism in <span class="SimpleMath">\(\mathrm{Hom}( s, r )\)</span></p>
<p>The arguments are an object <span class="SimpleMath">\(s = (a \otimes b_1) \oplus \dots \oplus (a \otimes b_n)\)</span>, an object <span class="SimpleMath">\(a\)</span>, a list of objects <span class="SimpleMath">\(L = (b_1, \dots, b_n)\)</span>, and an object <span class="SimpleMath">\(r = a \otimes (b_1 \oplus \dots \oplus b_n)\)</span>. The output is the left distributivity morphism <span class="SimpleMath">\(s \rightarrow r\)</span>.</p>
<p><a id="X8409C74778A4313A" name="X8409C74778A4313A"></a></p>
<h5>1.2-5 RightDistributivityExpanding</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ RightDistributivityExpanding</code>( <var class="Arg">L</var>, <var class="Arg">a</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: a morphism in <span class="SimpleMath">\(\mathrm{Hom}( (b_1 \oplus \dots \oplus b_n) \otimes a, (b_1 \otimes a) \oplus \dots \oplus (b_n \otimes a) )\)</span></p>
<p>The arguments are a list of objects <span class="SimpleMath">\(L = (b_1, \dots, b_n)\)</span> and an object <span class="SimpleMath">\(a\)</span>. The output is the right distributivity morphism <span class="SimpleMath">\((b_1 \oplus \dots \oplus b_n) \otimes a \rightarrow (b_1 \otimes a) \oplus \dots \oplus (b_n \otimes a)\)</span>.</p>
<p><a id="X7DBE629F811DEADC" name="X7DBE629F811DEADC"></a></p>
<h5>1.2-6 RightDistributivityExpandingWithGivenObjects</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ RightDistributivityExpandingWithGivenObjects</code>( <var class="Arg">s</var>, <var class="Arg">L</var>, <var class="Arg">a</var>, <var class="Arg">r</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: a morphism in <span class="SimpleMath">\(\mathrm{Hom}( s, r )\)</span></p>
<p>The arguments are an object <span class="SimpleMath">\(s = (b_1 \oplus \dots \oplus b_n) \otimes a\)</span>, a list of objects <span class="SimpleMath">\(L = (b_1, \dots, b_n)\)</span>, an object <span class="SimpleMath">\(a\)</span>, and an object <span class="SimpleMath">\(r = (b_1 \otimes a) \oplus \dots \oplus (b_n \otimes a)\)</span>. The output is the right distributivity morphism <span class="SimpleMath">\(s \rightarrow r\)</span>.</p>
<p><a id="X86C5C33786C1DC28" name="X86C5C33786C1DC28"></a></p>
<h5>1.2-7 RightDistributivityFactoring</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ RightDistributivityFactoring</code>( <var class="Arg">L</var>, <var class="Arg">a</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: a morphism in <span class="SimpleMath">\(\mathrm{Hom}( (b_1 \otimes a) \oplus \dots \oplus (b_n \otimes a), (b_1 \oplus \dots \oplus b_n) \otimes a)\)</span></p>
<p>The arguments are a list of objects <span class="SimpleMath">\(L = (b_1, \dots, b_n)\)</span> and an object <span class="SimpleMath">\(a\)</span>. The output is the right distributivity morphism <span class="SimpleMath">\((b_1 \otimes a) \oplus \dots \oplus (b_n \otimes a) \rightarrow (b_1 \oplus \dots \oplus b_n) \otimes a \)</span>.</p>
<p><a id="X795267217F7807CE" name="X795267217F7807CE"></a></p>
<h5>1.2-8 RightDistributivityFactoringWithGivenObjects</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ RightDistributivityFactoringWithGivenObjects</code>( <var class="Arg">s</var>, <var class="Arg">L</var>, <var class="Arg">a</var>, <var class="Arg">r</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: a morphism in <span class="SimpleMath">\(\mathrm{Hom}( s, r )\)</span></p>
<p>The arguments are an object <span class="SimpleMath">\(s = (b_1 \otimes a) \oplus \dots \oplus (b_n \otimes a)\)</span>, a list of objects <span class="SimpleMath">\(L = (b_1, \dots, b_n)\)</span>, an object <span class="SimpleMath">\(a\)</span>, and an object <span class="SimpleMath">\(r = (b_1 \oplus \dots \oplus b_n) \otimes a\)</span>. The output is the right distributivity morphism <span class="SimpleMath">\(s \rightarrow r\)</span>.</p>
<p><a id="X7D94AD967E6D60F1" name="X7D94AD967E6D60F1"></a></p>
<h4>1.3 <span class="Heading">Braided Monoidal Categories</span></h4>
<p>A monoidal category <span class="SimpleMath">\(\mathbf{C}\)</span> equipped with a natural isomorphism <span class="SimpleMath">\(B_{a,b}: a \otimes b \cong b \otimes a\)</span> is called a <em>braided monoidal category</em> if</p>
<ul>
<li><p><span class="SimpleMath">\(\lambda_a \circ B_{a,1} \sim \rho_a\)</span>,</p>
</li>
<li><p><span class="SimpleMath">\((B_{c,a} \otimes \mathrm{id}_b) \circ \alpha_{c,a,b} \circ B_{a \otimes b,c} \sim \alpha_{a,c,b} \circ ( \mathrm{id}_a \otimes B_{b,c}) \circ \alpha^{-1}_{a,b,c}\)</span>,</p>
</li>
<li><p><span class="SimpleMath">\(( \mathrm{id}_b \otimes B_{c,a} ) \circ \alpha^{-1}_{b,c,a} \circ B_{a,b \otimes c} \sim \alpha^{-1}_{b,a,c} \circ (B_{a,b} \otimes \mathrm{id}_c) \circ \alpha_{a,b,c}\)</span>.</p>
</li>
</ul>
<p>The corresponding GAP property is given by <code class="code">IsBraidedMonoidalCategory</code>.</p>
<p><a id="X7C6ADBFE7A8DD1E3" name="X7C6ADBFE7A8DD1E3"></a></p>
<h5>1.3-1 Braiding</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Braiding</code>( <var class="Arg">a</var>, <var class="Arg">b</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: a morphism in <span class="SimpleMath">\(\mathrm{Hom}( a \otimes b, b \otimes a )\)</span>.</p>
<p>The arguments are two objects <span class="SimpleMath">\(a,b\)</span>. The output is the braiding <span class="SimpleMath">\( B_{a,b}: a \otimes b \rightarrow b \otimes a\)</span>.</p>
<p><a id="X7F4B09A87B2500C0" name="X7F4B09A87B2500C0"></a></p>
<h5>1.3-2 BraidingWithGivenTensorProducts</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ BraidingWithGivenTensorProducts</code>( <var class="Arg">s</var>, <var class="Arg">a</var>, <var class="Arg">b</var>, <var class="Arg">r</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: a morphism in <span class="SimpleMath">\(\mathrm{Hom}( a \otimes b, b \otimes a )\)</span>.</p>
<p>The arguments are an object <span class="SimpleMath">\(s = a \otimes b\)</span>, two objects <span class="SimpleMath">\(a,b\)</span>, and an object <span class="SimpleMath">\(r = b \otimes a\)</span>. The output is the braiding <span class="SimpleMath">\( B_{a,b}: a \otimes b \rightarrow b \otimes a\)</span>.</p>
<p><a id="X8195EC5279D3E8D0" name="X8195EC5279D3E8D0"></a></p>
<h5>1.3-3 BraidingInverse</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ BraidingInverse</code>( <var class="Arg">a</var>, <var class="Arg">b</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: a morphism in <span class="SimpleMath">\(\mathrm{Hom}( b \otimes a, a \otimes b )\)</span>.</p>
<p>The arguments are two objects <span class="SimpleMath">\(a,b\)</span>. The output is the inverse braiding <span class="SimpleMath">\( B_{a,b}^{-1}: b \otimes a \rightarrow a \otimes b\)</span>.</p>
<p><a id="X854AFBC67C836769" name="X854AFBC67C836769"></a></p>
<h5>1.3-4 BraidingInverseWithGivenTensorProducts</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ BraidingInverseWithGivenTensorProducts</code>( <var class="Arg">s</var>, <var class="Arg">a</var>, <var class="Arg">b</var>, <var class="Arg">r</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: a morphism in <span class="SimpleMath">\(\mathrm{Hom}( b \otimes a, a \otimes b )\)</span>.</p>
<p>The arguments are an object <span class="SimpleMath">\(s = b \otimes a\)</span>, two objects <span class="SimpleMath">\(a,b\)</span>, and an object <span class="SimpleMath">\(r = a \otimes b\)</span>. The output is the inverse braiding <span class="SimpleMath">\( B_{a,b}^{-1}: b \otimes a \rightarrow a \otimes b\)</span>.</p>
<p><a id="X85ED71067F7CEA82" name="X85ED71067F7CEA82"></a></p>
<h4>1.4 <span class="Heading">Symmetric Monoidal Categories</span></h4>
<p>A braided monoidal category <span class="SimpleMath">\(\mathbf{C}\)</span> is called <em>symmetric monoidal category</em> if <span class="SimpleMath">\(B_{a,b}^{-1} \sim B_{b,a}\)</span>. The corresponding GAP property is given by <code class="code">IsSymmetricMonoidalCategory</code>.</p>
<p><a id="X85035E9683B050D0" name="X85035E9683B050D0"></a></p>
<h4>1.5 <span class="Heading">Left Closed Monoidal Categories</span></h4>
<p>A monoidal category <span class="SimpleMath">\(\mathbf{C}\)</span> which has for each functor <span class="SimpleMath">\(- \otimes b: \mathbf{C} \rightarrow \mathbf{C}\)</span> a right adjoint (denoted by <span class="SimpleMath">\(\mathrm{\underline{Hom}_\ell}(b,-)\)</span>) is called a <em>left closed monoidal category</em>.</p>
<p>If no operations involving left duals are installed manually, the left dual objects will be derived as <span class="SimpleMath">\(a^\vee \coloneqq \mathrm{\underline{Hom}_\ell}(a,1)\)</span>.</p>
<p>The corresponding GAP property is called <code class="code">IsLeftClosedMonoidalCategory</code>.</p>
<p><a id="X87A4C0637EB8A18E" name="X87A4C0637EB8A18E"></a></p>
<h5>1.5-1 LeftInternalHomOnObjects</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ LeftInternalHomOnObjects</code>( <var class="Arg">a</var>, <var class="Arg">b</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: an object</p>
<p>The arguments are two objects <span class="SimpleMath">\(a,b\)</span>. The output is the internal hom object <span class="SimpleMath">\(\mathrm{\underline{Hom}_\ell}(a,b)\)</span>.</p>
<p><a id="X8439FF407D2C9C6E" name="X8439FF407D2C9C6E"></a></p>
<h5>1.5-2 LeftInternalHomOnMorphisms</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ LeftInternalHomOnMorphisms</code>( <var class="Arg">alpha</var>, <var class="Arg">beta</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: a morphism in <span class="SimpleMath">\(\mathrm{Hom}( \mathrm{\underline{Hom}_\ell}(a',b), \mathrm{\underline{Hom}_\ell}(a,b') )\)</span></p>
<p>The arguments are two morphisms <span class="SimpleMath">\(\alpha: a \rightarrow a', \beta: b \rightarrow b'\)</span>. The output is the internal hom morphism <span class="SimpleMath">\(\mathrm{\underline{Hom}_\ell}(\alpha,\beta): \mathrm{\underline{Hom}_\ell}(a',b) \rightarrow \mathrm{\underline{Hom}_\ell}(a,b')\)</span>.</p>
<p><a id="X7861C2AD82F62C77" name="X7861C2AD82F62C77"></a></p>
<h5>1.5-3 LeftInternalHomOnMorphismsWithGivenLeftInternalHoms</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ LeftInternalHomOnMorphismsWithGivenLeftInternalHoms</code>( <var class="Arg">s</var>, <var class="Arg">alpha</var>, <var class="Arg">beta</var>, <var class="Arg">r</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: a morphism in <span class="SimpleMath">\(\mathrm{Hom}( s, r )\)</span></p>
<p>The arguments are an object <span class="SimpleMath">\(s = \mathrm{\underline{Hom}_\ell}(a',b)\)</span>, two morphisms <span class="SimpleMath">\(\alpha: a \rightarrow a', \beta: b \rightarrow b'\)</span>, and an object <span class="SimpleMath">\(r = \mathrm{\underline{Hom}_\ell}(a,b')\)</span>. The output is the internal hom morphism <span class="SimpleMath">\(\mathrm{\underline{Hom}_\ell}(\alpha,\beta): \mathrm{\underline{Hom}_\ell}(a',b) \rightarrow \mathrm{\underline{Hom}_\ell}(a,b')\)</span>.</p>
<p><a id="X82C4FACF83C3611D" name="X82C4FACF83C3611D"></a></p>
<h5>1.5-4 LeftClosedMonoidalEvaluationMorphism</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ LeftClosedMonoidalEvaluationMorphism</code>( <var class="Arg">a</var>, <var class="Arg">b</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: a morphism in <span class="SimpleMath">\(\mathrm{Hom}( \mathrm{\underline{Hom}_\ell}(a,b) \otimes a, b )\)</span>.</p>
<p>The arguments are two objects <span class="SimpleMath">\(a, b\)</span>. The output is the evaluation morphism <span class="SimpleMath">\(\mathrm{ev}_{a,b}: \mathrm{\underline{Hom}_\ell}(a,b) \otimes a \rightarrow b\)</span>, i.e., the counit of the tensor hom adjunction.</p>
<p><a id="X8575A3748713EACF" name="X8575A3748713EACF"></a></p>
<h5>1.5-5 LeftClosedMonoidalEvaluationMorphismWithGivenSource</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ LeftClosedMonoidalEvaluationMorphismWithGivenSource</code>( <var class="Arg">a</var>, <var class="Arg">b</var>, <var class="Arg">s</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: a morphism in <span class="SimpleMath">\(\mathrm{Hom}( s, b )\)</span>.</p>
<p>The arguments are two objects <span class="SimpleMath">\(a,b\)</span> and an object <span class="SimpleMath">\(s = \mathrm{\underline{Hom}_\ell}(a,b) \otimes a\)</span>. The output is the evaluation morphism <span class="SimpleMath">\(\mathrm{ev}_{a,b}: \mathrm{\underline{Hom}_\ell}(a,b) \otimes a \rightarrow b\)</span>, i.e., the counit of the tensor hom adjunction.</p>
<p><a id="X7A526C59812F2A59" name="X7A526C59812F2A59"></a></p>
<h5>1.5-6 LeftClosedMonoidalCoevaluationMorphism</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ LeftClosedMonoidalCoevaluationMorphism</code>( <var class="Arg">a</var>, <var class="Arg">b</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: a morphism in <span class="SimpleMath">\(\mathrm{Hom}( b, \mathrm{\underline{Hom}_\ell}(a, b \otimes a) )\)</span>.</p>
<p>The arguments are two objects <span class="SimpleMath">\(a,b\)</span>. The output is the coevaluation morphism <span class="SimpleMath">\(\mathrm{coev}_{a,b}: b \rightarrow \mathrm{\underline{Hom}_\ell}(a, b \otimes a)\)</span>, i.e., the unit of the tensor hom adjunction.</p>
<p><a id="X7BC7F1497A00E87A" name="X7BC7F1497A00E87A"></a></p>
<h5>1.5-7 LeftClosedMonoidalCoevaluationMorphismWithGivenRange</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ LeftClosedMonoidalCoevaluationMorphismWithGivenRange</code>( <var class="Arg">a</var>, <var class="Arg">b</var>, <var class="Arg">r</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: a morphism in <span class="SimpleMath">\(\mathrm{Hom}( b, r )\)</span>.</p>
<p>The arguments are two objects <span class="SimpleMath">\(a,b\)</span> and an object <span class="SimpleMath">\(r = \mathrm{\underline{Hom}_\ell}(a, b \otimes a)\)</span>. The output is the coevaluation morphism <span class="SimpleMath">\(\mathrm{coev}_{a,b}: b \rightarrow \mathrm{\underline{Hom}_\ell}(a, b \otimes a)\)</span>, i.e., the unit of the tensor hom adjunction.</p>
<p><a id="X83A5A47979C9B92D" name="X83A5A47979C9B92D"></a></p>
<h5>1.5-8 TensorProductToLeftInternalHomAdjunctMorphism</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ TensorProductToLeftInternalHomAdjunctMorphism</code>( <var class="Arg">a</var>, <var class="Arg">b</var>, <var class="Arg">f</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: a morphism in <span class="SimpleMath">\(\mathrm{Hom}( a, \mathrm{\underline{Hom}_\ell}(b,c) )\)</span>.</p>
<p>The arguments are two objects <span class="SimpleMath">\(a,b\)</span> and a morphism <span class="SimpleMath">\(f: a \otimes b \rightarrow c\)</span>. The output is a morphism <span class="SimpleMath">\(g: a \rightarrow \mathrm{\underline{Hom}_\ell}(b,c)\)</span> corresponding to <span class="SimpleMath">\(f\)</span> under the tensor hom adjunction.</p>
<p><a id="X7BB734AA7F213B47" name="X7BB734AA7F213B47"></a></p>
<h5>1.5-9 TensorProductToLeftInternalHomAdjunctMorphismWithGivenLeftInternalHom</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ TensorProductToLeftInternalHomAdjunctMorphismWithGivenLeftInternalHom</code>( <var class="Arg">a</var>, <var class="Arg">b</var>, <var class="Arg">f</var>, <var class="Arg">i</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: a morphism in <span class="SimpleMath">\(\mathrm{Hom}( a, i )\)</span>.</p>
<p>The arguments are two objects <span class="SimpleMath">\(a,b\)</span>, a morphism <span class="SimpleMath">\(f: a \otimes b \rightarrow c\)</span> and an object <span class="SimpleMath">\(i = \mathrm{\underline{Hom}_\ell}(b,c)\)</span>. The output is a morphism <span class="SimpleMath">\(g: a \rightarrow \mathrm{\underline{Hom}_\ell}(b,c)\)</span> corresponding to <span class="SimpleMath">\(f\)</span> under the tensor hom adjunction.</p>
<p><a id="X7C7D769879A7DA90" name="X7C7D769879A7DA90"></a></p>
<h5>1.5-10 LeftInternalHomToTensorProductAdjunctMorphism</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ LeftInternalHomToTensorProductAdjunctMorphism</code>( <var class="Arg">b</var>, <var class="Arg">c</var>, <var class="Arg">g</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: a morphism in <span class="SimpleMath">\(\mathrm{Hom}(a \otimes b, c)\)</span>.</p>
<p>The arguments are two objects <span class="SimpleMath">\(b,c\)</span> and a morphism <span class="SimpleMath">\(g: a \rightarrow \mathrm{\underline{Hom}_\ell}(b,c)\)</span>. The output is a morphism <span class="SimpleMath">\(f: a \otimes b \rightarrow c\)</span> corresponding to <span class="SimpleMath">\(g\)</span> under the tensor hom adjunction.</p>
<p><a id="X80431C73861AEC43" name="X80431C73861AEC43"></a></p>
<h5>1.5-11 LeftInternalHomToTensorProductAdjunctMorphismWithGivenTensorProduct</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ LeftInternalHomToTensorProductAdjunctMorphismWithGivenTensorProduct</code>( <var class="Arg">b</var>, <var class="Arg">c</var>, <var class="Arg">g</var>, <var class="Arg">t</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: a morphism in <span class="SimpleMath">\(\mathrm{Hom}(t, c)\)</span>.</p>
<p>The arguments are two objects <span class="SimpleMath">\(b,c\)</span>, a morphism <span class="SimpleMath">\(g: a \rightarrow \mathrm{\underline{Hom}_\ell}(b,c)\)</span> and an object <span class="SimpleMath">\(t = a \otimes b\)</span>. The output is a morphism <span class="SimpleMath">\(f: a \otimes b \rightarrow c\)</span> corresponding to <span class="SimpleMath">\(g\)</span> under the tensor hom adjunction.</p>
<p><a id="X79AC308B847F5177" name="X79AC308B847F5177"></a></p>
<h5>1.5-12 LeftClosedMonoidalPreComposeMorphism</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ LeftClosedMonoidalPreComposeMorphism</code>( <var class="Arg">a</var>, <var class="Arg">b</var>, <var class="Arg">c</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: a morphism in <span class="SimpleMath">\(\mathrm{Hom}( \mathrm{\underline{Hom}_\ell}(a,b) \otimes \mathrm{\underline{Hom}_\ell}(b,c), \mathrm{\underline{Hom}_\ell}(a,c) )\)</span>.</p>
<p>The arguments are three objects <span class="SimpleMath">\(a,b,c\)</span>. The output is the precomposition morphism <span class="SimpleMath">\(\mathrm{LeftClosedMonoidalPreComposeMorphism}_{a,b,c}: \mathrm{\underline{Hom}_\ell}(a,b) \otimes \mathrm{\underline{Hom}_\ell}(b,c) \rightarrow \mathrm{\underline{Hom}_\ell}(a,c)\)</span>.</p>
<p><a id="X78E3E67480AC81F4" name="X78E3E67480AC81F4"></a></p>
<h5>1.5-13 LeftClosedMonoidalPreComposeMorphismWithGivenObjects</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ LeftClosedMonoidalPreComposeMorphismWithGivenObjects</code>( <var class="Arg">s</var>, <var class="Arg">a</var>, <var class="Arg">b</var>, <var class="Arg">c</var>, <var class="Arg">r</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: a morphism in <span class="SimpleMath">\(\mathrm{Hom}( s, r )\)</span>.</p>
<p>The arguments are an object <span class="SimpleMath">\(s = \mathrm{\underline{Hom}_\ell}(a,b) \otimes \mathrm{\underline{Hom}_\ell}(b,c)\)</span>, three objects <span class="SimpleMath">\(a,b,c\)</span>, and an object <span class="SimpleMath">\(r = \mathrm{\underline{Hom}_\ell}(a,c)\)</span>. The output is the precomposition morphism <span class="SimpleMath">\(\mathrm{LeftClosedMonoidalPreComposeMorphismWithGivenObjects}_{a,b,c}: \mathrm{\underline{Hom}_\ell}(a,b) \otimes \mathrm{\underline{Hom}_\ell}(b,c) \rightarrow \mathrm{\underline{Hom}_\ell}(a,c)\)</span>.</p>
<p><a id="X7CFBE32D7F8F16CD" name="X7CFBE32D7F8F16CD"></a></p>
<h5>1.5-14 LeftClosedMonoidalPostComposeMorphism</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ LeftClosedMonoidalPostComposeMorphism</code>( <var class="Arg">a</var>, <var class="Arg">b</var>, <var class="Arg">c</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: a morphism in <span class="SimpleMath">\(\mathrm{Hom}( \mathrm{\underline{Hom}_\ell}(b,c) \otimes \mathrm{\underline{Hom}_\ell}(a,b), \mathrm{\underline{Hom}_\ell}(a,c) )\)</span>.</p>
<p>The arguments are three objects <span class="SimpleMath">\(a,b,c\)</span>. The output is the postcomposition morphism <span class="SimpleMath">\(\mathrm{LeftClosedMonoidalPostComposeMorphism}_{a,b,c}: \mathrm{\underline{Hom}_\ell}(b,c) \otimes \mathrm{\underline{Hom}_\ell}(a,b) \rightarrow \mathrm{\underline{Hom}_\ell}(a,c)\)</span>.</p>
<p><a id="X7D5E05257D908A56" name="X7D5E05257D908A56"></a></p>
<h5>1.5-15 LeftClosedMonoidalPostComposeMorphismWithGivenObjects</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ LeftClosedMonoidalPostComposeMorphismWithGivenObjects</code>( <var class="Arg">s</var>, <var class="Arg">a</var>, <var class="Arg">b</var>, <var class="Arg">c</var>, <var class="Arg">r</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: a morphism in <span class="SimpleMath">\(\mathrm{Hom}( s, r )\)</span>.</p>
<p>The arguments are an object <span class="SimpleMath">\(s = \mathrm{\underline{Hom}_\ell}(b,c) \otimes \mathrm{\underline{Hom}_\ell}(a,b)\)</span>, three objects <span class="SimpleMath">\(a,b,c\)</span>, and an object <span class="SimpleMath">\(r = \mathrm{\underline{Hom}_\ell}(a,c)\)</span>. The output is the postcomposition morphism <span class="SimpleMath">\(\mathrm{LeftClosedMonoidalPostComposeMorphismWithGivenObjects}_{a,b,c}: \mathrm{\underline{Hom}_\ell}(b,c) \otimes \mathrm{\underline{Hom}_\ell}(a,b) \rightarrow \mathrm{\underline{Hom}_\ell}(a,c)\)</span>.</p>
<p><a id="X859959ED7F2CFDCC" name="X859959ED7F2CFDCC"></a></p>
<h5>1.5-16 LeftDualOnObjects</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ LeftDualOnObjects</code>( <var class="Arg">a</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: an object</p>
<p>The argument is an object <span class="SimpleMath">\(a\)</span>. The output is its dual object <span class="SimpleMath">\(a^{\vee}\)</span>.</p>
<p><a id="X7B697FFC7B23A48B" name="X7B697FFC7B23A48B"></a></p>
<h5>1.5-17 LeftDualOnMorphisms</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ LeftDualOnMorphisms</code>( <var class="Arg">alpha</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: a morphism in <span class="SimpleMath">\(\mathrm{Hom}( b^{\vee}, a^{\vee} )\)</span>.</p>
<p>The argument is a morphism <span class="SimpleMath">\(\alpha: a \rightarrow b\)</span>. The output is its dual morphism <span class="SimpleMath">\(\alpha^{\vee}: b^{\vee} \rightarrow a^{\vee}\)</span>.</p>
<p><a id="X8282E454874316D8" name="X8282E454874316D8"></a></p>
<h5>1.5-18 LeftDualOnMorphismsWithGivenLeftDuals</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ LeftDualOnMorphismsWithGivenLeftDuals</code>( <var class="Arg">s</var>, <var class="Arg">alpha</var>, <var class="Arg">r</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: a morphism in <span class="SimpleMath">\(\mathrm{Hom}( s, r )\)</span>.</p>
<p>The argument is an object <span class="SimpleMath">\(s = b^{\vee}\)</span>, a morphism <span class="SimpleMath">\(\alpha: a \rightarrow b\)</span>, and an object <span class="SimpleMath">\(r = a^{\vee}\)</span>. The output is the dual morphism <span class="SimpleMath">\(\alpha^{\vee}: b^{\vee} \rightarrow a^{\vee}\)</span>.</p>
<p><a id="X7FF3227087ABDA8E" name="X7FF3227087ABDA8E"></a></p>
<h5>1.5-19 LeftClosedMonoidalEvaluationForLeftDual</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ LeftClosedMonoidalEvaluationForLeftDual</code>( <var class="Arg">a</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: a morphism in <span class="SimpleMath">\(\mathrm{Hom}( a^{\vee} \otimes a, 1 )\)</span>.</p>
<p>The argument is an object <span class="SimpleMath">\(a\)</span>. The output is the evaluation morphism <span class="SimpleMath">\(\mathrm{ev}_{a}: a^{\vee} \otimes a \rightarrow 1\)</span>.</p>
<p><a id="X86995E4D7A63E68F" name="X86995E4D7A63E68F"></a></p>
<h5>1.5-20 LeftClosedMonoidalEvaluationForLeftDualWithGivenTensorProduct</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ LeftClosedMonoidalEvaluationForLeftDualWithGivenTensorProduct</code>( <var class="Arg">s</var>, <var class="Arg">a</var>, <var class="Arg">r</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: a morphism in <span class="SimpleMath">\(\mathrm{Hom}( s, r )\)</span>.</p>
<p>The arguments are an object <span class="SimpleMath">\(s = a^{\vee} \otimes a\)</span>, an object <span class="SimpleMath">\(a\)</span>, and an object <span class="SimpleMath">\(r = 1\)</span>. The output is the evaluation morphism <span class="SimpleMath">\(\mathrm{ev}_{a}: a^{\vee} \otimes a \rightarrow 1\)</span>.</p>
<p><a id="X85D2B94A85E3DBD5" name="X85D2B94A85E3DBD5"></a></p>
<h5>1.5-21 MorphismToLeftBidual</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ MorphismToLeftBidual</code>( <var class="Arg">a</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: a morphism in <span class="SimpleMath">\(\mathrm{Hom}(a, (a^{\vee})^{\vee})\)</span>.</p>
<p>The argument is an object <span class="SimpleMath">\(a\)</span>. The output is the morphism to the bidual <span class="SimpleMath">\(a \rightarrow (a^{\vee})^{\vee}\)</span>.</p>
<p><a id="X868774B68189D7E9" name="X868774B68189D7E9"></a></p>
<h5>1.5-22 MorphismToLeftBidualWithGivenLeftBidual</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ MorphismToLeftBidualWithGivenLeftBidual</code>( <var class="Arg">a</var>, <var class="Arg">r</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: a morphism in <span class="SimpleMath">\(\mathrm{Hom}(a, r)\)</span>.</p>
<p>The arguments are an object <span class="SimpleMath">\(a\)</span>, and an object <span class="SimpleMath">\(r = (a^{\vee})^{\vee}\)</span>. The output is the morphism to the bidual <span class="SimpleMath">\(a \rightarrow (a^{\vee})^{\vee}\)</span>.</p>
<p><a id="X7939CEE8854017D2" name="X7939CEE8854017D2"></a></p>
<h5>1.5-23 TensorProductLeftInternalHomCompatibilityMorphism</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ TensorProductLeftInternalHomCompatibilityMorphism</code>( <var class="Arg">list</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: a morphism in <span class="SimpleMath">\(\mathrm{Hom}( \mathrm{\underline{Hom}_\ell}(a,a') \otimes \mathrm{\underline{Hom}_\ell}(b,b'), \mathrm{\underline{Hom}_\ell}(a \otimes b,a' \otimes b'))\)</span>.</p>
<p>The argument is a list of four objects <span class="SimpleMath">\([ a, a', b, b' ]\)</span>. The ou | | |
