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59 lines
32 KiB
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<html><head><meta charset="utf-8"><title>Categories.Functor.Bifunctor.Properties</title><link rel="stylesheet" href="Agda.css"></head><body><pre class="Agda"><a id="1" class="Symbol">{-#</a> <a id="5" class="Keyword">OPTIONS</a> <a id="13" class="Pragma">--without-K</a> <a id="25" class="Pragma">--safe</a> <a id="32" class="Symbol">#-}</a>
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<a id="37" class="Keyword">module</a> <a id="44" href="Categories.Functor.Bifunctor.Properties.html" class="Module">Categories.Functor.Bifunctor.Properties</a> <a id="84" class="Keyword">where</a>
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<a id="91" class="Keyword">open</a> <a id="96" class="Keyword">import</a> <a id="103" href="Level.html" class="Module">Level</a>
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<a id="109" class="Keyword">open</a> <a id="114" class="Keyword">import</a> <a id="121" href="Data.Product.html" class="Module">Data.Product</a> <a id="134" class="Keyword">using</a> <a id="140" class="Symbol">(</a><a id="141" href="Agda.Builtin.Sigma.html#165" class="Record">Σ</a><a id="142" class="Symbol">;</a> <a id="144" href="Agda.Builtin.Sigma.html#235" class="InductiveConstructor Operator">_,_</a><a id="147" class="Symbol">)</a>
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<a id="150" class="Keyword">open</a> <a id="155" class="Keyword">import</a> <a id="162" href="Categories.Category.html" class="Module">Categories.Category</a>
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<a id="182" class="Keyword">open</a> <a id="187" class="Keyword">import</a> <a id="194" href="Categories.Functor.html" class="Module">Categories.Functor</a>
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<a id="213" class="Keyword">open</a> <a id="218" class="Keyword">import</a> <a id="225" href="Categories.Functor.Bifunctor.html" class="Module">Categories.Functor.Bifunctor</a>
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<a id="254" class="Keyword">import</a> <a id="261" href="Categories.Morphism.Reasoning.html" class="Module">Categories.Morphism.Reasoning</a> <a id="291" class="Symbol">as</a> <a id="294" class="Module">MR</a>
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<a id="298" class="Keyword">private</a>
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<a id="308" class="Keyword">variable</a>
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<a id="321" href="Categories.Functor.Bifunctor.Properties.html#321" class="Generalizable">o</a> <a id="323" href="Categories.Functor.Bifunctor.Properties.html#323" class="Generalizable">ℓ</a> <a id="325" href="Categories.Functor.Bifunctor.Properties.html#325" class="Generalizable">e</a> <a id="327" class="Symbol">:</a> <a id="329" href="Agda.Primitive.html#742" class="Postulate">Level</a>
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<a id="339" href="Categories.Functor.Bifunctor.Properties.html#339" class="Generalizable">C</a> <a id="341" href="Categories.Functor.Bifunctor.Properties.html#341" class="Generalizable">D</a> <a id="343" href="Categories.Functor.Bifunctor.Properties.html#343" class="Generalizable">E</a> <a id="345" class="Symbol">:</a> <a id="347" href="Categories.Category.Core.html#442" class="Record">Category</a> <a id="356" href="Categories.Functor.Bifunctor.Properties.html#321" class="Generalizable">o</a> <a id="358" href="Categories.Functor.Bifunctor.Properties.html#323" class="Generalizable">ℓ</a> <a id="360" href="Categories.Functor.Bifunctor.Properties.html#325" class="Generalizable">e</a>
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<a id="363" class="Keyword">module</a> <a id="370" href="Categories.Functor.Bifunctor.Properties.html#370" class="Module">_</a> <a id="372" class="Symbol">(</a><a id="373" href="Categories.Functor.Bifunctor.Properties.html#373" class="Bound">F</a> <a id="375" class="Symbol">:</a> <a id="377" href="Categories.Functor.Bifunctor.html#441" class="Function">Bifunctor</a> <a id="387" href="Categories.Functor.Bifunctor.Properties.html#339" class="Generalizable">C</a> <a id="389" href="Categories.Functor.Bifunctor.Properties.html#341" class="Generalizable">D</a> <a id="391" href="Categories.Functor.Bifunctor.Properties.html#343" class="Generalizable">E</a><a id="392" class="Symbol">)</a> <a id="394" class="Keyword">where</a>
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<a id="402" class="Keyword">private</a>
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<a id="414" class="Keyword">module</a> <a id="421" href="Categories.Functor.Bifunctor.Properties.html#421" class="Module">C</a> <a id="423" class="Symbol">=</a> <a id="425" href="Categories.Category.Core.html#442" class="Module">Category</a> <a id="434" href="Categories.Functor.Bifunctor.Properties.html#387" class="Bound">C</a>
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<a id="440" class="Keyword">module</a> <a id="447" href="Categories.Functor.Bifunctor.Properties.html#447" class="Module">D</a> <a id="449" class="Symbol">=</a> <a id="451" href="Categories.Category.Core.html#442" class="Module">Category</a> <a id="460" href="Categories.Functor.Bifunctor.Properties.html#389" class="Bound">D</a>
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<a id="466" class="Keyword">module</a> <a id="473" href="Categories.Functor.Bifunctor.Properties.html#473" class="Module">E</a> <a id="475" class="Symbol">=</a> <a id="477" href="Categories.Category.Core.html#442" class="Module">Category</a> <a id="486" href="Categories.Functor.Bifunctor.Properties.html#391" class="Bound">E</a>
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<a id="493" class="Keyword">variable</a>
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<a id="508" href="Categories.Functor.Bifunctor.Properties.html#508" class="Generalizable">A</a> <a id="510" href="Categories.Functor.Bifunctor.Properties.html#510" class="Generalizable">B</a> <a id="512" class="Symbol">:</a> <a id="514" href="Categories.Category.Core.html#559" class="Function">C.Obj</a>
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<a id="526" href="Categories.Functor.Bifunctor.Properties.html#526" class="Generalizable">X</a> <a id="528" href="Categories.Functor.Bifunctor.Properties.html#528" class="Generalizable">Y</a> <a id="530" class="Symbol">:</a> <a id="532" href="Categories.Category.Core.html#559" class="Function">D.Obj</a>
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<a id="544" href="Categories.Functor.Bifunctor.Properties.html#544" class="Generalizable">f</a> <a id="546" href="Categories.Functor.Bifunctor.Properties.html#546" class="Generalizable">h</a> <a id="548" href="Categories.Functor.Bifunctor.Properties.html#548" class="Generalizable">j</a> <a id="550" class="Symbol">:</a> <a id="552" href="Categories.Functor.Bifunctor.Properties.html#508" class="Generalizable">A</a> <a id="554" href="Categories.Category.Core.html#575" class="Function Operator">C.⇒</a> <a id="558" href="Categories.Functor.Bifunctor.Properties.html#510" class="Generalizable">B</a>
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<a id="566" href="Categories.Functor.Bifunctor.Properties.html#566" class="Generalizable">g</a> <a id="568" href="Categories.Functor.Bifunctor.Properties.html#568" class="Generalizable">i</a> <a id="570" href="Categories.Functor.Bifunctor.Properties.html#570" class="Generalizable">k</a> <a id="572" class="Symbol">:</a> <a id="574" href="Categories.Functor.Bifunctor.Properties.html#526" class="Generalizable">X</a> <a id="576" href="Categories.Category.Core.html#575" class="Function Operator">D.⇒</a> <a id="580" href="Categories.Functor.Bifunctor.Properties.html#528" class="Generalizable">Y</a>
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<a id="585" class="Keyword">open</a> <a id="590" href="Categories.Functor.Bifunctor.Properties.html#473" class="Module">E</a>
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<a id="594" class="Keyword">open</a> <a id="599" href="Categories.Category.Core.html#2462" class="Module">HomReasoning</a>
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<a id="614" class="Keyword">open</a> <a id="619" href="Categories.Functor.Core.html#248" class="Module">Functor</a> <a id="627" href="Categories.Functor.Bifunctor.Properties.html#373" class="Bound">F</a>
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<a id="632" href="Categories.Functor.Bifunctor.Properties.html#632" class="Function Operator">[_]-commute</a> <a id="644" class="Symbol">:</a> <a id="646" href="Categories.Functor.Core.html#455" class="Field">F₁</a> <a id="649" class="Symbol">(</a><a id="650" href="Categories.Category.Core.html#630" class="Function">C.id</a> <a id="655" href="Agda.Builtin.Sigma.html#235" class="InductiveConstructor Operator">,</a> <a id="657" href="Categories.Functor.Bifunctor.Properties.html#566" class="Generalizable">g</a><a id="658" class="Symbol">)</a> <a id="660" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="662" href="Categories.Functor.Core.html#455" class="Field">F₁</a> <a id="665" class="Symbol">(</a><a id="666" href="Categories.Functor.Bifunctor.Properties.html#544" class="Generalizable">f</a> <a id="668" href="Agda.Builtin.Sigma.html#235" class="InductiveConstructor Operator">,</a> <a id="670" href="Categories.Category.Core.html#630" class="Function">D.id</a><a id="674" class="Symbol">)</a> <a id="676" href="Categories.Category.Core.html#595" class="Function Operator">≈</a> <a id="678" href="Categories.Functor.Core.html#455" class="Field">F₁</a> <a id="681" class="Symbol">(</a><a id="682" href="Categories.Functor.Bifunctor.Properties.html#544" class="Generalizable">f</a> <a id="684" href="Agda.Builtin.Sigma.html#235" class="InductiveConstructor Operator">,</a> <a id="686" href="Categories.Category.Core.html#630" class="Function">D.id</a><a id="690" class="Symbol">)</a> <a id="692" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="694" href="Categories.Functor.Core.html#455" class="Field">F₁</a> <a id="697" class="Symbol">(</a><a id="698" href="Categories.Category.Core.html#630" class="Function">C.id</a> <a id="703" href="Agda.Builtin.Sigma.html#235" class="InductiveConstructor Operator">,</a> <a id="705" href="Categories.Functor.Bifunctor.Properties.html#566" class="Generalizable">g</a><a id="706" class="Symbol">)</a>
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<a id="710" href="Categories.Functor.Bifunctor.Properties.html#632" class="Function Operator">[_]-commute</a> <a id="722" class="Symbol">{</a><a id="723" class="Argument">g</a> <a id="725" class="Symbol">=</a> <a id="727" href="Categories.Functor.Bifunctor.Properties.html#727" class="Bound">g</a><a id="728" class="Symbol">}</a> <a id="730" class="Symbol">{</a><a id="731" class="Argument">f</a> <a id="733" class="Symbol">=</a> <a id="735" href="Categories.Functor.Bifunctor.Properties.html#735" class="Bound">f</a><a id="736" class="Symbol">}</a> <a id="738" class="Symbol">=</a> <a id="740" href="Relation.Binary.Reasoning.Syntax.html#1510" class="Function Operator">begin</a>
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<a id="750" href="Categories.Functor.Core.html#455" class="Field">F₁</a> <a id="753" class="Symbol">(</a><a id="754" href="Categories.Category.Core.html#630" class="Function">C.id</a> <a id="759" href="Agda.Builtin.Sigma.html#235" class="InductiveConstructor Operator">,</a> <a id="761" href="Categories.Functor.Bifunctor.Properties.html#727" class="Bound">g</a><a id="762" class="Symbol">)</a> <a id="764" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="766" href="Categories.Functor.Core.html#455" class="Field">F₁</a> <a id="769" class="Symbol">(</a><a id="770" href="Categories.Functor.Bifunctor.Properties.html#735" class="Bound">f</a> <a id="772" href="Agda.Builtin.Sigma.html#235" class="InductiveConstructor Operator">,</a> <a id="774" href="Categories.Category.Core.html#630" class="Function">D.id</a><a id="778" class="Symbol">)</a> <a id="780" href="Relation.Binary.Reasoning.Syntax.html#7400" class="Function">≈˘⟨</a> <a id="784" href="Categories.Functor.Core.html#565" class="Field">homomorphism</a> <a id="797" href="Relation.Binary.Reasoning.Syntax.html#7400" class="Function">⟩</a>
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<a id="803" href="Categories.Functor.Core.html#455" class="Field">F₁</a> <a id="806" class="Symbol">(</a><a id="807" href="Categories.Category.Core.html#630" class="Function">C.id</a> <a id="812" href="Categories.Category.Core.html#656" class="Function Operator">C.∘</a> <a id="816" href="Categories.Functor.Bifunctor.Properties.html#735" class="Bound">f</a> <a id="818" href="Agda.Builtin.Sigma.html#235" class="InductiveConstructor Operator">,</a> <a id="820" href="Categories.Functor.Bifunctor.Properties.html#727" class="Bound">g</a> <a id="822" href="Categories.Category.Core.html#656" class="Function Operator">D.∘</a> <a id="826" href="Categories.Category.Core.html#630" class="Function">D.id</a><a id="830" class="Symbol">)</a> <a id="833" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="836" href="Categories.Functor.Core.html#696" class="Field">F-resp-≈</a> <a id="845" class="Symbol">(</a><a id="846" href="Categories.Morphism.Reasoning.Core.html#1309" class="Function">MR.id-comm-sym</a> <a id="861" href="Categories.Functor.Bifunctor.Properties.html#387" class="Bound">C</a> <a id="863" href="Agda.Builtin.Sigma.html#235" class="InductiveConstructor Operator">,</a> <a id="865" href="Categories.Morphism.Reasoning.Core.html#1202" class="Function">MR.id-comm</a> <a id="876" href="Categories.Functor.Bifunctor.Properties.html#389" class="Bound">D</a><a id="877" class="Symbol">)</a> <a id="879" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
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<a id="885" href="Categories.Functor.Core.html#455" class="Field">F₁</a> <a id="888" class="Symbol">(</a><a id="889" href="Categories.Functor.Bifunctor.Properties.html#735" class="Bound">f</a> <a id="891" href="Categories.Category.Core.html#656" class="Function Operator">C.∘</a> <a id="895" href="Categories.Category.Core.html#630" class="Function">C.id</a> <a id="900" href="Agda.Builtin.Sigma.html#235" class="InductiveConstructor Operator">,</a> <a id="902" href="Categories.Category.Core.html#630" class="Function">D.id</a> <a id="907" href="Categories.Category.Core.html#656" class="Function Operator">D.∘</a> <a id="911" href="Categories.Functor.Bifunctor.Properties.html#727" class="Bound">g</a><a id="912" class="Symbol">)</a> <a id="915" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="918" href="Categories.Functor.Core.html#565" class="Field">homomorphism</a> <a id="931" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
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<a id="937" href="Categories.Functor.Core.html#455" class="Field">F₁</a> <a id="940" class="Symbol">(</a><a id="941" href="Categories.Functor.Bifunctor.Properties.html#735" class="Bound">f</a> <a id="943" href="Agda.Builtin.Sigma.html#235" class="InductiveConstructor Operator">,</a> <a id="945" href="Categories.Category.Core.html#630" class="Function">D.id</a><a id="949" class="Symbol">)</a> <a id="951" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="953" href="Categories.Functor.Core.html#455" class="Field">F₁</a> <a id="956" class="Symbol">(</a><a id="957" href="Categories.Category.Core.html#630" class="Function">C.id</a> <a id="962" href="Agda.Builtin.Sigma.html#235" class="InductiveConstructor Operator">,</a> <a id="964" href="Categories.Functor.Bifunctor.Properties.html#727" class="Bound">g</a><a id="965" class="Symbol">)</a> <a id="967" href="Relation.Binary.Reasoning.Syntax.html#12283" class="Function Operator">∎</a>
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<a id="972" href="Categories.Functor.Bifunctor.Properties.html#972" class="Function Operator">[_]-decompose₁</a> <a id="987" class="Symbol">:</a> <a id="989" href="Categories.Functor.Core.html#455" class="Field">F₁</a> <a id="992" class="Symbol">(</a><a id="993" href="Categories.Functor.Bifunctor.Properties.html#544" class="Generalizable">f</a> <a id="995" href="Agda.Builtin.Sigma.html#235" class="InductiveConstructor Operator">,</a> <a id="997" href="Categories.Functor.Bifunctor.Properties.html#566" class="Generalizable">g</a><a id="998" class="Symbol">)</a> <a id="1000" href="Categories.Category.Core.html#595" class="Function Operator">≈</a> <a id="1002" href="Categories.Functor.Core.html#455" class="Field">F₁</a> <a id="1005" class="Symbol">(</a><a id="1006" href="Categories.Functor.Bifunctor.Properties.html#544" class="Generalizable">f</a> <a id="1008" href="Agda.Builtin.Sigma.html#235" class="InductiveConstructor Operator">,</a> <a id="1010" href="Categories.Category.Core.html#630" class="Function">D.id</a><a id="1014" class="Symbol">)</a> <a id="1016" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="1018" href="Categories.Functor.Core.html#455" class="Field">F₁</a> <a id="1021" class="Symbol">(</a><a id="1022" href="Categories.Category.Core.html#630" class="Function">C.id</a> <a id="1027" href="Agda.Builtin.Sigma.html#235" class="InductiveConstructor Operator">,</a> <a id="1029" href="Categories.Functor.Bifunctor.Properties.html#566" class="Generalizable">g</a><a id="1030" class="Symbol">)</a>
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<a id="1034" href="Categories.Functor.Bifunctor.Properties.html#972" class="Function Operator">[_]-decompose₁</a> <a id="1049" class="Symbol">{</a><a id="1050" class="Argument">f</a> <a id="1052" class="Symbol">=</a> <a id="1054" href="Categories.Functor.Bifunctor.Properties.html#1054" class="Bound">f</a><a id="1055" class="Symbol">}</a> <a id="1057" class="Symbol">{</a><a id="1058" class="Argument">g</a> <a id="1060" class="Symbol">=</a> <a id="1062" href="Categories.Functor.Bifunctor.Properties.html#1062" class="Bound">g</a><a id="1063" class="Symbol">}</a> <a id="1065" class="Symbol">=</a> <a id="1067" href="Relation.Binary.Reasoning.Syntax.html#1510" class="Function Operator">begin</a>
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<a id="1077" href="Categories.Functor.Core.html#455" class="Field">F₁</a> <a id="1080" class="Symbol">(</a><a id="1081" href="Categories.Functor.Bifunctor.Properties.html#1054" class="Bound">f</a> <a id="1083" href="Agda.Builtin.Sigma.html#235" class="InductiveConstructor Operator">,</a> <a id="1085" href="Categories.Functor.Bifunctor.Properties.html#1062" class="Bound">g</a><a id="1086" class="Symbol">)</a> <a id="1107" href="Relation.Binary.Reasoning.Syntax.html#7400" class="Function">≈˘⟨</a> <a id="1111" href="Categories.Functor.Core.html#696" class="Field">F-resp-≈</a> <a id="1120" class="Symbol">(</a><a id="1121" href="Categories.Category.Core.html#1145" class="Function">C.identityʳ</a> <a id="1133" href="Agda.Builtin.Sigma.html#235" class="InductiveConstructor Operator">,</a> <a id="1135" href="Categories.Category.Core.html#1096" class="Function">D.identityˡ</a><a id="1146" class="Symbol">)</a> <a id="1148" href="Relation.Binary.Reasoning.Syntax.html#7400" class="Function">⟩</a>
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<a id="1154" href="Categories.Functor.Core.html#455" class="Field">F₁</a> <a id="1157" class="Symbol">(</a><a id="1158" href="Categories.Functor.Bifunctor.Properties.html#1054" class="Bound">f</a> <a id="1160" href="Categories.Category.Core.html#656" class="Function Operator">C.∘</a> <a id="1164" href="Categories.Category.Core.html#630" class="Function">C.id</a> <a id="1169" href="Agda.Builtin.Sigma.html#235" class="InductiveConstructor Operator">,</a> <a id="1171" href="Categories.Category.Core.html#630" class="Function">D.id</a> <a id="1176" href="Categories.Category.Core.html#656" class="Function Operator">D.∘</a> <a id="1180" href="Categories.Functor.Bifunctor.Properties.html#1062" class="Bound">g</a><a id="1181" class="Symbol">)</a> <a id="1184" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="1187" href="Categories.Functor.Core.html#565" class="Field">homomorphism</a> <a id="1200" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
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<a id="1206" href="Categories.Functor.Core.html#455" class="Field">F₁</a> <a id="1209" class="Symbol">(</a><a id="1210" href="Categories.Functor.Bifunctor.Properties.html#1054" class="Bound">f</a> <a id="1212" href="Agda.Builtin.Sigma.html#235" class="InductiveConstructor Operator">,</a> <a id="1214" href="Categories.Category.Core.html#630" class="Function">D.id</a><a id="1218" class="Symbol">)</a> <a id="1220" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="1222" href="Categories.Functor.Core.html#455" class="Field">F₁</a> <a id="1225" class="Symbol">(</a><a id="1226" href="Categories.Category.Core.html#630" class="Function">C.id</a> <a id="1231" href="Agda.Builtin.Sigma.html#235" class="InductiveConstructor Operator">,</a> <a id="1233" href="Categories.Functor.Bifunctor.Properties.html#1062" class="Bound">g</a><a id="1234" class="Symbol">)</a> <a id="1236" href="Relation.Binary.Reasoning.Syntax.html#12283" class="Function Operator">∎</a>
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<a id="1241" href="Categories.Functor.Bifunctor.Properties.html#1241" class="Function Operator">[_]-decompose₂</a> <a id="1256" class="Symbol">:</a> <a id="1258" href="Categories.Functor.Core.html#455" class="Field">F₁</a> <a id="1261" class="Symbol">(</a><a id="1262" href="Categories.Functor.Bifunctor.Properties.html#544" class="Generalizable">f</a> <a id="1264" href="Agda.Builtin.Sigma.html#235" class="InductiveConstructor Operator">,</a> <a id="1266" href="Categories.Functor.Bifunctor.Properties.html#566" class="Generalizable">g</a><a id="1267" class="Symbol">)</a> <a id="1269" href="Categories.Category.Core.html#595" class="Function Operator">≈</a> <a id="1271" href="Categories.Functor.Core.html#455" class="Field">F₁</a> <a id="1274" class="Symbol">(</a><a id="1275" href="Categories.Category.Core.html#630" class="Function">C.id</a> <a id="1280" href="Agda.Builtin.Sigma.html#235" class="InductiveConstructor Operator">,</a> <a id="1282" href="Categories.Functor.Bifunctor.Properties.html#566" class="Generalizable">g</a><a id="1283" class="Symbol">)</a> <a id="1285" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="1287" href="Categories.Functor.Core.html#455" class="Field">F₁</a> <a id="1290" class="Symbol">(</a><a id="1291" href="Categories.Functor.Bifunctor.Properties.html#544" class="Generalizable">f</a> <a id="1293" href="Agda.Builtin.Sigma.html#235" class="InductiveConstructor Operator">,</a> <a id="1295" href="Categories.Category.Core.html#630" class="Function">D.id</a><a id="1299" class="Symbol">)</a>
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<a id="1303" href="Categories.Functor.Bifunctor.Properties.html#1241" class="Function Operator">[_]-decompose₂</a> <a id="1318" class="Symbol">{</a><a id="1319" class="Argument">f</a> <a id="1321" class="Symbol">=</a> <a id="1323" href="Categories.Functor.Bifunctor.Properties.html#1323" class="Bound">f</a><a id="1324" class="Symbol">}</a> <a id="1326" class="Symbol">{</a><a id="1327" class="Argument">g</a> <a id="1329" class="Symbol">=</a> <a id="1331" href="Categories.Functor.Bifunctor.Properties.html#1331" class="Bound">g</a><a id="1332" class="Symbol">}</a> <a id="1334" class="Symbol">=</a> <a id="1336" href="Relation.Binary.Reasoning.Syntax.html#1510" class="Function Operator">begin</a>
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<a id="1346" href="Categories.Functor.Core.html#455" class="Field">F₁</a> <a id="1349" class="Symbol">(</a><a id="1350" href="Categories.Functor.Bifunctor.Properties.html#1323" class="Bound">f</a> <a id="1352" href="Agda.Builtin.Sigma.html#235" class="InductiveConstructor Operator">,</a> <a id="1354" href="Categories.Functor.Bifunctor.Properties.html#1331" class="Bound">g</a><a id="1355" class="Symbol">)</a> <a id="1376" href="Relation.Binary.Reasoning.Syntax.html#7400" class="Function">≈˘⟨</a> <a id="1380" href="Categories.Functor.Core.html#696" class="Field">F-resp-≈</a> <a id="1389" class="Symbol">(</a><a id="1390" href="Categories.Category.Core.html#1096" class="Function">C.identityˡ</a> <a id="1402" href="Agda.Builtin.Sigma.html#235" class="InductiveConstructor Operator">,</a> <a id="1404" href="Categories.Category.Core.html#1145" class="Function">D.identityʳ</a><a id="1415" class="Symbol">)</a> <a id="1417" href="Relation.Binary.Reasoning.Syntax.html#7400" class="Function">⟩</a>
|
|||
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|
<a id="1423" href="Categories.Functor.Core.html#455" class="Field">F₁</a> <a id="1426" class="Symbol">(</a><a id="1427" href="Categories.Category.Core.html#630" class="Function">C.id</a> <a id="1432" href="Categories.Category.Core.html#656" class="Function Operator">C.∘</a> <a id="1436" href="Categories.Functor.Bifunctor.Properties.html#1323" class="Bound">f</a> <a id="1438" href="Agda.Builtin.Sigma.html#235" class="InductiveConstructor Operator">,</a> <a id="1440" href="Categories.Functor.Bifunctor.Properties.html#1331" class="Bound">g</a> <a id="1442" href="Categories.Category.Core.html#656" class="Function Operator">D.∘</a> <a id="1446" href="Categories.Category.Core.html#630" class="Function">D.id</a><a id="1450" class="Symbol">)</a> <a id="1453" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="1456" href="Categories.Functor.Core.html#565" class="Field">homomorphism</a> <a id="1469" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
|
|||
|
|
<a id="1475" href="Categories.Functor.Core.html#455" class="Field">F₁</a> <a id="1478" class="Symbol">(</a><a id="1479" href="Categories.Category.Core.html#630" class="Function">C.id</a> <a id="1484" href="Agda.Builtin.Sigma.html#235" class="InductiveConstructor Operator">,</a> <a id="1486" href="Categories.Functor.Bifunctor.Properties.html#1331" class="Bound">g</a><a id="1487" class="Symbol">)</a> <a id="1489" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="1491" href="Categories.Functor.Core.html#455" class="Field">F₁</a> <a id="1494" class="Symbol">(</a><a id="1495" href="Categories.Functor.Bifunctor.Properties.html#1323" class="Bound">f</a> <a id="1497" href="Agda.Builtin.Sigma.html#235" class="InductiveConstructor Operator">,</a> <a id="1499" href="Categories.Category.Core.html#630" class="Function">D.id</a><a id="1503" class="Symbol">)</a> <a id="1505" href="Relation.Binary.Reasoning.Syntax.html#12283" class="Function Operator">∎</a>
|
|||
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|
|||
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<a id="1510" href="Categories.Functor.Bifunctor.Properties.html#1510" class="Function Operator">[_]-merge</a> <a id="1520" class="Symbol">:</a> <a id="1522" href="Categories.Functor.Bifunctor.Properties.html#544" class="Generalizable">f</a> <a id="1524" href="Categories.Category.Core.html#656" class="Function Operator">C.∘</a> <a id="1528" href="Categories.Functor.Bifunctor.Properties.html#546" class="Generalizable">h</a> <a id="1530" href="Categories.Category.Core.html#595" class="Function Operator">C.≈</a> <a id="1534" href="Categories.Functor.Bifunctor.Properties.html#548" class="Generalizable">j</a> <a id="1536" class="Symbol">→</a> <a id="1538" href="Categories.Functor.Bifunctor.Properties.html#566" class="Generalizable">g</a> <a id="1540" href="Categories.Category.Core.html#656" class="Function Operator">D.∘</a> <a id="1544" href="Categories.Functor.Bifunctor.Properties.html#568" class="Generalizable">i</a> <a id="1546" href="Categories.Category.Core.html#595" class="Function Operator">D.≈</a> <a id="1550" href="Categories.Functor.Bifunctor.Properties.html#570" class="Generalizable">k</a> <a id="1552" class="Symbol">→</a> <a id="1555" href="Categories.Functor.Core.html#455" class="Field">F₁</a> <a id="1558" class="Symbol">(</a><a id="1559" href="Categories.Functor.Bifunctor.Properties.html#544" class="Generalizable">f</a> <a id="1561" href="Agda.Builtin.Sigma.html#235" class="InductiveConstructor Operator">,</a> <a id="1563" href="Categories.Functor.Bifunctor.Properties.html#566" class="Generalizable">g</a><a id="1564" class="Symbol">)</a> <a id="1566" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="1568" href="Categories.Functor.Core.html#455" class="Field">F₁</a> <a id="1571" class="Symbol">(</a><a id="1572" href="Categories.Functor.Bifunctor.Properties.html#546" class="Generalizable">h</a> <a id="1574" href="Agda.Builtin.Sigma.html#235" class="InductiveConstructor Operator">,</a> <a id="1576" href="Categories.Functor.Bifunctor.Properties.html#568" class="Generalizable">i</a><a id="1577" class="Symbol">)</a> <a id="1579" href="Categories.Category.Core.html#595" class="Function Operator">≈</a> <a id="1581" href="Categories.Functor.Core.html#455" class="Field">F₁</a> <a id="1584" class="Symbol">(</a><a id="1585" href="Categories.Functor.Bifunctor.Properties.html#548" class="Generalizable">j</a> <a id="1587" href="Agda.Builtin.Sigma.html#235" class="InductiveConstructor Operator">,</a> <a id="1589" href="Categories.Functor.Bifunctor.Properties.html#570" class="Generalizable">k</a><a id="1590" class="Symbol">)</a>
|
|||
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<a id="1594" href="Categories.Functor.Bifunctor.Properties.html#1510" class="Function Operator">[_]-merge</a> <a id="1604" class="Symbol">{</a><a id="1605" class="Argument">f</a> <a id="1607" class="Symbol">=</a> <a id="1609" href="Categories.Functor.Bifunctor.Properties.html#1609" class="Bound">f</a><a id="1610" class="Symbol">}</a> <a id="1612" class="Symbol">{</a><a id="1613" class="Argument">h</a> <a id="1615" class="Symbol">=</a> <a id="1617" href="Categories.Functor.Bifunctor.Properties.html#1617" class="Bound">h</a><a id="1618" class="Symbol">}</a> <a id="1620" class="Symbol">{</a><a id="1621" class="Argument">j</a> <a id="1623" class="Symbol">=</a> <a id="1625" href="Categories.Functor.Bifunctor.Properties.html#1625" class="Bound">j</a><a id="1626" class="Symbol">}</a> <a id="1628" class="Symbol">{</a><a id="1629" class="Argument">g</a> <a id="1631" class="Symbol">=</a> <a id="1633" href="Categories.Functor.Bifunctor.Properties.html#1633" class="Bound">g</a><a id="1634" class="Symbol">}</a> <a id="1636" class="Symbol">{</a><a id="1637" class="Argument">i</a> <a id="1639" class="Symbol">=</a> <a id="1641" href="Categories.Functor.Bifunctor.Properties.html#1641" class="Bound">i</a><a id="1642" class="Symbol">}</a> <a id="1644" class="Symbol">{</a><a id="1645" class="Argument">k</a> <a id="1647" class="Symbol">=</a> <a id="1649" href="Categories.Functor.Bifunctor.Properties.html#1649" class="Bound">k</a><a id="1650" class="Symbol">}</a> <a id="1652" href="Categories.Functor.Bifunctor.Properties.html#1652" class="Bound">eq</a> <a id="1655" href="Categories.Functor.Bifunctor.Properties.html#1655" class="Bound">eq′</a> <a id="1659" class="Symbol">=</a> <a id="1661" href="Relation.Binary.Reasoning.Syntax.html#1510" class="Function Operator">begin</a>
|
|||
|
|
<a id="1671" href="Categories.Functor.Core.html#455" class="Field">F₁</a> <a id="1674" class="Symbol">(</a><a id="1675" href="Categories.Functor.Bifunctor.Properties.html#1609" class="Bound">f</a> <a id="1677" href="Agda.Builtin.Sigma.html#235" class="InductiveConstructor Operator">,</a> <a id="1679" href="Categories.Functor.Bifunctor.Properties.html#1633" class="Bound">g</a><a id="1680" class="Symbol">)</a> <a id="1682" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="1684" href="Categories.Functor.Core.html#455" class="Field">F₁</a> <a id="1687" class="Symbol">(</a><a id="1688" href="Categories.Functor.Bifunctor.Properties.html#1617" class="Bound">h</a> <a id="1690" href="Agda.Builtin.Sigma.html#235" class="InductiveConstructor Operator">,</a> <a id="1692" href="Categories.Functor.Bifunctor.Properties.html#1641" class="Bound">i</a><a id="1693" class="Symbol">)</a> <a id="1695" href="Relation.Binary.Reasoning.Syntax.html#7400" class="Function">≈˘⟨</a> <a id="1699" href="Categories.Functor.Core.html#565" class="Field">homomorphism</a> <a id="1712" href="Relation.Binary.Reasoning.Syntax.html#7400" class="Function">⟩</a>
|
|||
|
|
<a id="1718" href="Categories.Functor.Core.html#455" class="Field">F₁</a> <a id="1721" class="Symbol">(</a><a id="1722" href="Categories.Functor.Bifunctor.Properties.html#1609" class="Bound">f</a> <a id="1724" href="Categories.Category.Core.html#656" class="Function Operator">C.∘</a> <a id="1728" href="Categories.Functor.Bifunctor.Properties.html#1617" class="Bound">h</a> <a id="1730" href="Agda.Builtin.Sigma.html#235" class="InductiveConstructor Operator">,</a> <a id="1732" href="Categories.Functor.Bifunctor.Properties.html#1633" class="Bound">g</a> <a id="1734" href="Categories.Category.Core.html#656" class="Function Operator">D.∘</a> <a id="1738" href="Categories.Functor.Bifunctor.Properties.html#1641" class="Bound">i</a><a id="1739" class="Symbol">)</a> <a id="1742" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="1745" href="Categories.Functor.Core.html#696" class="Field">F-resp-≈</a> <a id="1754" class="Symbol">(</a><a id="1755" href="Categories.Functor.Bifunctor.Properties.html#1652" class="Bound">eq</a> <a id="1758" href="Agda.Builtin.Sigma.html#235" class="InductiveConstructor Operator">,</a> <a id="1760" href="Categories.Functor.Bifunctor.Properties.html#1655" class="Bound">eq′</a><a id="1763" class="Symbol">)</a> <a id="1765" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
|
|||
|
|
<a id="1771" href="Categories.Functor.Core.html#455" class="Field">F₁</a> <a id="1774" class="Symbol">(</a><a id="1775" href="Categories.Functor.Bifunctor.Properties.html#1625" class="Bound">j</a> <a id="1777" href="Agda.Builtin.Sigma.html#235" class="InductiveConstructor Operator">,</a> <a id="1779" href="Categories.Functor.Bifunctor.Properties.html#1649" class="Bound">k</a><a id="1780" class="Symbol">)</a> <a id="1795" href="Relation.Binary.Reasoning.Syntax.html#12283" class="Function Operator">∎</a>
|
|||
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</pre></body></html>
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