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<h1>MonoidalCategories</h1>
<h2>Monoidal and monoidal (co)closed categories</h2>
<p>
2025.08-02</p>
<p>
25 August 2025
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<p><b>
Mohamed Barakat
</b>
<br />Email: <span class="URL"><a href="mailto:mohamed.barakat@uni-siegen.de">mohamed.barakat@uni-siegen.de</a></span>
<br />Homepage: <span class="URL"><a href="http://algebra.mathematik.uni-siegen.de/barakat/">http://algebra.mathematik.uni-siegen.de/barakat/</a></span>
<br />Address: <br />Walter-Flex-Str. 3<br /> 57068 Siegen<br /> Germany<br />
</p><p><b>
Sebastian Gutsche
</b>
<br />Email: <span class="URL"><a href="mailto:sebastian.gutsche@gmail.com">sebastian.gutsche@gmail.com</a></span>
<br />Homepage: <span class="URL"><a href="http://algebra.mathematik.uni-siegen.de/gutsche/">http://algebra.mathematik.uni-siegen.de/gutsche/</a></span>
<br />Address: <br />Department Mathematik<br /> Universität Siegen<br /> Walter-Flex-Straße 3<br /> 57068 Siegen<br /> Germany<br />
</p><p><b>
Sebastian Posur
</b>
<br />Email: <span class="URL"><a href="mailto:sebastian.posur@uni-muenster.de">sebastian.posur@uni-muenster.de</a></span>
<br />Homepage: <span class="URL"><a href="http://algebra.mathematik.uni-siegen.de/posur/">http://algebra.mathematik.uni-siegen.de/posur/</a></span>
<br />Address: <br />Department Mathematik<br /> Universität Siegen<br /> Walter-Flex-Straße 3<br /> 57068 Siegen<br /> Germany<br />
</p><p><b>
Tom Kuhmichel
</b>
<br />Email: <span class="URL"><a href="mailto:tom.kuhmichel@student.uni-siegen.de">tom.kuhmichel@student.uni-siegen.de</a></span>
<br />Homepage: <span class="URL"><a href="https://github.com/TKuh">https://github.com/TKuh</a></span>
<br />Address: <br />Department Mathematik<br /> Universität Siegen<br /> Walter-Flex-Straße 3<br /> 57068 Siegen<br /> Germany<br />
</p><p><b>
Fabian Zickgraf
</b>
<br />Email: <span class="URL"><a href="mailto:f.zickgraf@dashdos.com">f.zickgraf@dashdos.com</a></span>
<br />Homepage: <span class="URL"><a href="https://github.com/zickgraf/">https://github.com/zickgraf/</a></span>
<br />Address: <br />Walter-Flex-Str. 3<br /> 57068 Siegen<br /> Germany<br />
</p>
<p><a id="X8537FEB07AF2BEC8" name="X8537FEB07AF2BEC8"></a></p>
<div class="contents">
<h3>Contents<a id="contents" name="contents"></a></h3>
<div class="ContChap"><a href="chap1.html#X808DC49C7ED99B52">1 <span class="Heading">Monoidal Categories</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap1.html#X808DC49C7ED99B52">1.1 <span class="Heading">Monoidal Categories</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X84997A1E8188D6BE">1.1-1 TensorProductOnMorphisms</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X85008BF07C2C0386">1.1-2 TensorProductOnMorphismsWithGivenTensorProducts</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X84C2FAEA7B9678AA">1.1-3 AssociatorRightToLeft</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X85645C4D86060ED2">1.1-4 AssociatorRightToLeftWithGivenTensorProducts</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X7C5A064C7E1995F1">1.1-5 AssociatorLeftToRight</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X7C59D73F8389E389">1.1-6 AssociatorLeftToRightWithGivenTensorProducts</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X7EBCAFDD81BDB655">1.1-7 LeftUnitor</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X7ED284DF7BA21774">1.1-8 LeftUnitorWithGivenTensorProduct</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X79A274078147A9F7">1.1-9 LeftUnitorInverse</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X849609447CAE996B">1.1-10 LeftUnitorInverseWithGivenTensorProduct</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X82F758CB7A23468A">1.1-11 RightUnitor</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X84C8F97F7BF2282D">1.1-12 RightUnitorWithGivenTensorProduct</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X82DC825480E87301">1.1-13 RightUnitorInverse</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X85A8A87C87B33008">1.1-14 RightUnitorInverseWithGivenTensorProduct</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X797034F17AA47EE0">1.1-15 TensorProductOnObjects</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X7BF7FFDF8789474A">1.1-16 TensorUnit</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap1.html#X7FB8551A815BEC53">1.2 <span class="Heading">Additive Monoidal Categories</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X7ACF268E7D82B0C9">1.2-1 LeftDistributivityExpanding</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X7CA18968862D2285">1.2-2 LeftDistributivityExpandingWithGivenObjects</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X780322FE7E5A626D">1.2-3 LeftDistributivityFactoring</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X784D8CD685F5F021">1.2-4 LeftDistributivityFactoringWithGivenObjects</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X8409C74778A4313A">1.2-5 RightDistributivityExpanding</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X7DBE629F811DEADC">1.2-6 RightDistributivityExpandingWithGivenObjects</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X86C5C33786C1DC28">1.2-7 RightDistributivityFactoring</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X795267217F7807CE">1.2-8 RightDistributivityFactoringWithGivenObjects</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap1.html#X7D94AD967E6D60F1">1.3 <span class="Heading">Braided Monoidal Categories</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X7C6ADBFE7A8DD1E3">1.3-1 Braiding</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X7F4B09A87B2500C0">1.3-2 BraidingWithGivenTensorProducts</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X8195EC5279D3E8D0">1.3-3 BraidingInverse</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X854AFBC67C836769">1.3-4 BraidingInverseWithGivenTensorProducts</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap1.html#X85ED71067F7CEA82">1.4 <span class="Heading">Symmetric Monoidal Categories</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap1.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.html#X87A4C0637EB8A18E">1.5-1 LeftInternalHomOnObjects</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X8439FF407D2C9C6E">1.5-2 LeftInternalHomOnMorphisms</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X7861C2AD82F62C77">1.5-3 LeftInternalHomOnMorphismsWithGivenLeftInternalHoms</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X82C4FACF83C3611D">1.5-4 LeftClosedMonoidalEvaluationMorphism</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X8575A3748713EACF">1.5-5 LeftClosedMonoidalEvaluationMorphismWithGivenSource</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X7A526C59812F2A59">1.5-6 LeftClosedMonoidalCoevaluationMorphism</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X7BC7F1497A00E87A">1.5-7 LeftClosedMonoidalCoevaluationMorphismWithGivenRange</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X83A5A47979C9B92D">1.5-8 TensorProductToLeftInternalHomAdjunctMorphism</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X7BB734AA7F213B47">1.5-9 TensorProductToLeftInternalHomAdjunctMorphismWithGivenLeftInternalHom</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X7C7D769879A7DA90">1.5-10 LeftInternalHomToTensorProductAdjunctMorphism</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X80431C73861AEC43">1.5-11 LeftInternalHomToTensorProductAdjunctMorphismWithGivenTensorProduct</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X79AC308B847F5177">1.5-12 LeftClosedMonoidalPreComposeMorphism</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X78E3E67480AC81F4">1.5-13 LeftClosedMonoidalPreComposeMorphismWithGivenObjects</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X7CFBE32D7F8F16CD">1.5-14 LeftClosedMonoidalPostComposeMorphism</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X7D5E05257D908A56">1.5-15 LeftClosedMonoidalPostComposeMorphismWithGivenObjects</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X859959ED7F2CFDCC">1.5-16 LeftDualOnObjects</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X7B697FFC7B23A48B">1.5-17 LeftDualOnMorphisms</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X8282E454874316D8">1.5-18 LeftDualOnMorphismsWithGivenLeftDuals</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X7FF3227087ABDA8E">1.5-19 LeftClosedMonoidalEvaluationForLeftDual</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X86995E4D7A63E68F">1.5-20 LeftClosedMonoidalEvaluationForLeftDualWithGivenTensorProduct</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X85D2B94A85E3DBD5">1.5-21 MorphismToLeftBidual</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X868774B68189D7E9">1.5-22 MorphismToLeftBidualWithGivenLeftBidual</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X7939CEE8854017D2">1.5-23 TensorProductLeftInternalHomCompatibilityMorphism</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X798C160A84CA5623">1.5-24 TensorProductLeftInternalHomCompatibilityMorphismWithGivenObjects</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X8635D47E7835C8AA">1.5-25 TensorProductLeftDualityCompatibilityMorphism</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X8435799E83221130">1.5-26 TensorProductLeftDualityCompatibilityMorphismWithGivenObjects</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X80FE157E81B02827">1.5-27 MorphismFromTensorProductToLeftInternalHom</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X78FDDC2986E50825">1.5-28 MorphismFromTensorProductToLeftInternalHomWithGivenObjects</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X7EA425DA848670D9">1.5-29 IsomorphismFromLeftDualObjectToLeftInternalHomIntoTensorUnit</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X781A2D417A8BC8DF">1.5-30 IsomorphismFromLeftInternalHomIntoTensorUnitToLeftDualObject</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X82D9D78C7A8DA61A">1.5-31 UniversalPropertyOfLeftDual</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X7C7A252A78484873">1.5-32 LeftClosedMonoidalLambdaIntroduction</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X83B3A8FA8363F620">1.5-33 LeftClosedMonoidalLambdaElimination</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X7ECAB43778388C41">1.5-34 IsomorphismFromObjectToLeftInternalHom</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X874D7D8E81A6AFF1">1.5-35 IsomorphismFromObjectToLeftInternalHomWithGivenLeftInternalHom</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X7908CF267DDAA38C">1.5-36 IsomorphismFromLeftInternalHomToObject</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X7B2F235C8444803C">1.5-37 IsomorphismFromLeftInternalHomToObjectWithGivenLeftInternalHom</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap1.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.html#X78AA83E77B380D68">1.6-1 InternalHomOnObjects</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X7C0191A483A72F98">1.6-2 InternalHomOnMorphisms</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X861226B47B5CB713">1.6-3 InternalHomOnMorphismsWithGivenInternalHoms</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X844F05DF7DE23F99">1.6-4 ClosedMonoidalRightEvaluationMorphism</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X83E6461280B0EB0C">1.6-5 ClosedMonoidalRightEvaluationMorphismWithGivenSource</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X7A745E5A836B5E91">1.6-6 ClosedMonoidalRightCoevaluationMorphism</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X868AD1A278EBAB03">1.6-7 ClosedMonoidalRightCoevaluationMorphismWithGivenRange</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X7BA9C4A184346304">1.6-8 TensorProductToInternalHomRightAdjunctMorphism</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X8218B01287CD44E5">1.6-9 TensorProductToInternalHomRightAdjunctMorphismWithGivenInternalHom</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X78A16762823E5F1C">1.6-10 TensorProductToInternalHomRightAdjunctionIsomorphism</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X81D37FA282FC8F11">1.6-11 TensorProductToInternalHomRightAdjunctionIsomorphismWithGivenObjects</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X8271B2767A2DFE76">1.6-12 InternalHomToTensorProductRightAdjunctMorphism</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X7ACBB0CE7FA4C781">1.6-13 InternalHomToTensorProductRightAdjunctMorphismWithGivenTensorProduct</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X79EEE4757C27C26E">1.6-14 InternalHomToTensorProductRightAdjunctionIsomorphism</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X7B4CD74A7CE51263">1.6-15 InternalHomToTensorProductRightAdjunctionIsomorphismWithGivenObjects</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X80B3C53A854B515B">1.6-16 ClosedMonoidalLeftEvaluationMorphism</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X84E1E136819BDA89">1.6-17 ClosedMonoidalLeftEvaluationMorphismWithGivenSource</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X7E35DEB787A71A1F">1.6-18 ClosedMonoidalLeftCoevaluationMorphism</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X8196CE5D7C88D83C">1.6-19 ClosedMonoidalLeftCoevaluationMorphismWithGivenRange</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X80A58B1F80A186BD">1.6-20 TensorProductToInternalHomLeftAdjunctMorphism</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X7CCE373482D60A57">1.6-21 TensorProductToInternalHomLeftAdjunctMorphismWithGivenInternalHom</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X793BAA50819573E4">1.6-22 TensorProductToInternalHomLeftAdjunctionIsomorphism</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X791276987824CDF8">1.6-23 TensorProductToInternalHomLeftAdjunctionIsomorphismWithGivenObjects</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X85F426677EB81BCF">1.6-24 InternalHomToTensorProductLeftAdjunctMorphism</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X859B738881052D1C">1.6-25 InternalHomToTensorProductLeftAdjunctMorphismWithGivenTensorProduct</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X86AD79F17F8CEE96">1.6-26 InternalHomToTensorProductLeftAdjunctionIsomorphism</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X7D707B26863D508A">1.6-27 InternalHomToTensorProductLeftAdjunctionIsomorphismWithGivenObjects</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X86E4E97C82BFB45C">1.6-28 MonoidalPreComposeMorphism</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X7FE4A52284060F20">1.6-29 MonoidalPreComposeMorphismWithGivenObjects</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X79F4FCE781385829">1.6-30 MonoidalPostComposeMorphism</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X7E6933B8816EE4E3">1.6-31 MonoidalPostComposeMorphismWithGivenObjects</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X80FFC71D7E57DD53">1.6-32 DualOnObjects</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X815C0BC47D000819">1.6-33 DualOnMorphisms</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X7DCC0F468386AA46">1.6-34 DualOnMorphismsWithGivenDuals</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X86D42C7587F03A68">1.6-35 EvaluationForDual</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X7B588B1B8472834E">1.6-36 EvaluationForDualWithGivenTensorProduct</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X7F68B55781C1DFB1">1.6-37 MorphismToBidual</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X79260E5680F1E741">1.6-38 MorphismToBidualWithGivenBidual</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X80CDA0CB821E08EE">1.6-39 TensorProductInternalHomCompatibilityMorphism</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X7B0194A07AB73486">1.6-40 TensorProductInternalHomCompatibilityMorphismWithGivenObjects</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X7D173DB08132E40A">1.6-41 TensorProductDualityCompatibilityMorphism</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X794ED97782B8DF76">1.6-42 TensorProductDualityCompatibilityMorphismWithGivenObjects</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X8439BFF57BFE390F">1.6-43 MorphismFromTensorProductToInternalHom</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X8473D28B7F3A5E20">1.6-44 MorphismFromTensorProductToInternalHomWithGivenObjects</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X8430726C82D6CFF7">1.6-45 IsomorphismFromDualObjectToInternalHomIntoTensorUnit</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X7A015736812A97C6">1.6-46 IsomorphismFromInternalHomIntoTensorUnitToDualObject</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X7AF8F68887146C20">1.6-47 UniversalPropertyOfDual</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X80F3F2287B9E55E3">1.6-48 LambdaIntroduction</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X79EC44077F661E80">1.6-49 LambdaElimination</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X8726BA888787D53D">1.6-50 IsomorphismFromObjectToInternalHom</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X87F91EF37C76A7F7">1.6-51 IsomorphismFromObjectToInternalHomWithGivenInternalHom</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X831CF4E1824FA178">1.6-52 IsomorphismFromInternalHomToObject</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X7988DC7C79BED3B2">1.6-53 IsomorphismFromInternalHomToObjectWithGivenInternalHom</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap1.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.html#X7CE5D10B7F10E9CC">1.7-1 LeftInternalCoHomOnObjects</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X853213767BB48099">1.7-2 LeftInternalCoHomOnMorphisms</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X7F302E3880752A2F">1.7-3 LeftInternalCoHomOnMorphismsWithGivenLeftInternalCoHoms</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X819A762D83B1C294">1.7-4 LeftCoclosedMonoidalEvaluationMorphism</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X7E5607B1789E00B7">1.7-5 LeftCoclosedMonoidalEvaluationMorphismWithGivenRange</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X81A249CE84828A9D">1.7-6 LeftCoclosedMonoidalCoevaluationMorphism</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X7D1018A284F21140">1.7-7 LeftCoclosedMonoidalCoevaluationMorphismWithGivenSource</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X851216B2807B3900">1.7-8 TensorProductToLeftInternalCoHomAdjunctMorphism</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X87BFA0FA7AFF2868">1.7-9 TensorProductToLeftInternalCoHomAdjunctMorphismWithGivenLeftInternalCoHom</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X7DDDACD58454C3A4">1.7-10 LeftInternalCoHomToTensorProductAdjunctMorphism</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X821AC96981CA794C">1.7-11 LeftInternalCoHomToTensorProductAdjunctMorphismWithGivenTensorProduct</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X8416357982DCBEE4">1.7-12 LeftCoclosedMonoidalPreCoComposeMorphism</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X7EECDE4583E14A59">1.7-13 LeftCoclosedMonoidalPreCoComposeMorphismWithGivenObjects</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X80A270E982F5BC2D">1.7-14 LeftCoclosedMonoidalPostCoComposeMorphism</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X867C6F087D72AB3F">1.7-15 LeftCoclosedMonoidalPostCoComposeMorphismWithGivenObjects</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X85A63AA57F0678EE">1.7-16 LeftCoDualOnObjects</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X7949DE307D2D5083">1.7-17 LeftCoDualOnMorphisms</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X84C835CB7E708596">1.7-18 LeftCoDualOnMorphismsWithGivenLeftCoDuals</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X7A20AC6A837F79BA">1.7-19 LeftCoclosedMonoidalEvaluationForLeftCoDual</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X87B15AC678955783">1.7-20 LeftCoclosedMonoidalEvaluationForLeftCoDualWithGivenTensorProduct</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X7F67676A85A47BDB">1.7-21 MorphismFromLeftCoBidual</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X7AF5F77180E09AAE">1.7-22 MorphismFromLeftCoBidualWithGivenLeftCoBidual</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X84331A1B7F62782F">1.7-23 LeftInternalCoHomTensorProductCompatibilityMorphism</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X808BEB0879D58CAF">1.7-24 LeftInternalCoHomTensorProductCompatibilityMorphismWithGivenObjects</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X7B4A66A8803C99BA">1.7-25 LeftCoDualityTensorProductCompatibilityMorphism</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X86E72092788A8F2D">1.7-26 LeftCoDualityTensorProductCompatibilityMorphismWithGivenObjects</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X7A5FDE4B83C5F328">1.7-27 MorphismFromLeftInternalCoHomToTensorProduct</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X7BD2A33F84A2FBAB">1.7-28 MorphismFromLeftInternalCoHomToTensorProductWithGivenObjects</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X7CC10C9E7EA7EED4">1.7-29 IsomorphismFromLeftCoDualObjectToLeftInternalCoHomFromTensorUnit</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X87F96B047F320A17">1.7-30 IsomorphismFromLeftInternalCoHomFromTensorUnitToLeftCoDualObject</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X7A0DE1667FF28457">1.7-31 UniversalPropertyOfLeftCoDual</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X85960F7479FFA8DA">1.7-32 LeftCoclosedMonoidalLambdaIntroduction</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X8721DDEC7DF9D89D">1.7-33 LeftCoclosedMonoidalLambdaElimination</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X853342457CC3B489">1.7-34 IsomorphismFromObjectToLeftInternalCoHom</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X82709BED807CB6FC">1.7-35 IsomorphismFromObjectToLeftInternalCoHomWithGivenLeftInternalCoHom</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X806C843B82108148">1.7-36 IsomorphismFromLeftInternalCoHomToObject</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X8370164B7EAF833D">1.7-37 IsomorphismFromLeftInternalCoHomToObjectWithGivenLeftInternalCoHom</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap1.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.html#X791BE0B0847A430D">1.8-1 InternalCoHomOnObjects</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X7FDF105D7F56EB53">1.8-2 InternalCoHomOnMorphisms</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X84DDA3DD878162B9">1.8-3 InternalCoHomOnMorphismsWithGivenInternalCoHoms</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X83CEC88D79AE5684">1.8-4 CoclosedMonoidalRightEvaluationMorphism</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X79333EC9822EA316">1.8-5 CoclosedMonoidalRightEvaluationMorphismWithGivenRange</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X80CA11547F3909E3">1.8-6 CoclosedMonoidalRightCoevaluationMorphism</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X7F1C10897EC4D14E">1.8-7 CoclosedMonoidalRightCoevaluationMorphismWithGivenSource</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X82B0937678F04730">1.8-8 TensorProductToInternalCoHomRightAdjunctMorphism</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X833D7B567810D7F3">1.8-9 TensorProductToInternalCoHomRightAdjunctMorphismWithGivenInternalCoHom</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X8623CC567D687FFD">1.8-10 InternalCoHomToTensorProductRightAdjunctMorphism</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X821B5212835BD1F6">1.8-11 InternalCoHomToTensorProductRightAdjunctMorphismWithGivenTensorProduct</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X81020D9E85C3280A">1.8-12 CoclosedMonoidalLeftEvaluationMorphism</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X7A1801D77EECEA29">1.8-13 CoclosedMonoidalLeftEvaluationMorphismWithGivenRange</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X7D6905F582F06003">1.8-14 CoclosedMonoidalLeftCoevaluationMorphism</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X802A3FC48280FBDE">1.8-15 CoclosedMonoidalLeftCoevaluationMorphismWithGivenSource</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X7FB4953B7F813D0B">1.8-16 TensorProductToInternalCoHomLeftAdjunctMorphism</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X8312DB0A82A97D4B">1.8-17 TensorProductToInternalCoHomLeftAdjunctMorphismWithGivenInternalCoHom</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X7A7FC1957A1905C6">1.8-18 InternalCoHomToTensorProductLeftAdjunctMorphism</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X85F970897F87BF2E">1.8-19 InternalCoHomToTensorProductLeftAdjunctMorphismWithGivenTensorProduct</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X8337DE6B7D61EAC2">1.8-20 MonoidalPreCoComposeMorphism</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X82A183B0807A0A01">1.8-21 MonoidalPreCoComposeMorphismWithGivenObjects</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X83FE0B8D853A76C1">1.8-22 MonoidalPostCoComposeMorphism</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X805DF38880B1E9E0">1.8-23 MonoidalPostCoComposeMorphismWithGivenObjects</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X817A1F7986256461">1.8-24 CoDualOnObjects</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X840CBC837926138E">1.8-25 CoDualOnMorphisms</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X79750BFD824D2AAF">1.8-26 CoDualOnMorphismsWithGivenCoDuals</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X7C65C05A788415E4">1.8-27 CoclosedEvaluationForCoDual</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X7CAB615C86D97CE8">1.8-28 CoclosedEvaluationForCoDualWithGivenTensorProduct</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X7841B6757A510799">1.8-29 MorphismFromCoBidual</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X7B2B002979E1CBF6">1.8-30 MorphismFromCoBidualWithGivenCoBidual</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X7D9E57197C820E0E">1.8-31 InternalCoHomTensorProductCompatibilityMorphism</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X81E9DC8D7FCD361F">1.8-32 InternalCoHomTensorProductCompatibilityMorphismWithGivenObjects</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X7B817BC97C1F1DF8">1.8-33 CoDualityTensorProductCompatibilityMorphism</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X82119ED686D3874C">1.8-34 CoDualityTensorProductCompatibilityMorphismWithGivenObjects</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X7A4971267B80B14F">1.8-35 MorphismFromInternalCoHomToTensorProduct</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X7B60FB927C37E125">1.8-36 MorphismFromInternalCoHomToTensorProductWithGivenObjects</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X86C7E40C85318EFA">1.8-37 IsomorphismFromCoDualObjectToInternalCoHomFromTensorUnit</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X84C86E29800BB8BC">1.8-38 IsomorphismFromInternalCoHomFromTensorUnitToCoDualObject</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X7AEE412D80799D09">1.8-39 UniversalPropertyOfCoDual</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X784CBFB984E66E7A">1.8-40 CoLambdaIntroduction</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X83520B098068CF62">1.8-41 CoLambdaElimination</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X83F11F6184DBD507">1.8-42 IsomorphismFromObjectToInternalCoHom</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X8233C995828387E9">1.8-43 IsomorphismFromObjectToInternalCoHomWithGivenInternalCoHom</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X875A349280E2095C">1.8-44 IsomorphismFromInternalCoHomToObject</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X858AD1F986BA5BB2">1.8-45 IsomorphismFromInternalCoHomToObjectWithGivenInternalCoHom</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap1.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.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.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.html#X782B629D7E7835C9">1.11-1 IsomorphismFromTensorProductWithDualObjectToInternalHom</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X84D0668C7CA0B63D">1.11-2 IsomorphismFromInternalHomToTensorProductWithDualObject</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X7D189B8280CECBA2">1.11-3 MorphismFromInternalHomToTensorProduct</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X7D10805D840AAC8D">1.11-4 MorphismFromInternalHomToTensorProductWithGivenObjects</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X85BA8C10817296F7">1.11-5 TensorProductInternalHomCompatibilityMorphismInverse</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X7CCE67B281EF45C1">1.11-6 TensorProductInternalHomCompatibilityMorphismInverseWithGivenObjects</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X83849B327C8074E9">1.11-7 CoevaluationForDual</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X7C65E6A97AAE0DE3">1.11-8 CoevaluationForDualWithGivenTensorProduct</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X85D0C9487A22AFFE">1.11-9 TraceMap</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X82F0DBD485D93793">1.11-10 RankMorphism</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X7E02E8197EA201EA">1.11-11 MorphismFromBidual</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X785CF0BB7BC0AC0D">1.11-12 MorphismFromBidualWithGivenBidual</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap1.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.html#X82D4EF587F1C194C">1.12-1 IsomorphismFromInternalCoHomToTensorProductWithCoDualObject</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X7FE736BA834D228A">1.12-2 IsomorphismFromTensorProductWithCoDualObjectToInternalCoHom</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X7DCA54857A6B45DF">1.12-3 MorphismFromTensorProductToInternalCoHom</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X7888F9947DDC15B5">1.12-4 MorphismFromTensorProductToInternalCoHomWithGivenObjects</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X7E71AF0985C5AEC5">1.12-5 InternalCoHomTensorProductCompatibilityMorphismInverse</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X87BF8BE67AD5ABCF">1.12-6 InternalCoHomTensorProductCompatibilityMorphismInverseWithGivenObjects</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X8695F9D97A8C6C61">1.12-7 CoclosedCoevaluationForCoDual</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X78B2703A7B81E340">1.12-8 CoclosedCoevaluationForCoDualWithGivenTensorProduct</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X829BE5F97A656200">1.12-9 CoTraceMap</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X7C5CC8F97D95AD43">1.12-10 CoRankMorphism</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X85BDB94D85C67725">1.12-11 MorphismToCoBidual</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X800035BF867418D0">1.12-12 MorphismToCoBidualWithGivenCoBidual</a></span>
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