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bsc-leon-vatthauer/agda/bsc-thesis/Categories.Category.Monoidal.Braided.Properties.html
Leon Vatthauer 2bf4655eae
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2024-02-09 17:53:52 +01:00

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<html><head><meta charset="utf-8"><title>Categories.Category.Monoidal.Braided.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>
<a id="37" class="Keyword">open</a> <a id="42" class="Keyword">import</a> <a id="49" href="Categories.Category.html" class="Module">Categories.Category</a> <a id="69" class="Keyword">using</a> <a id="75" class="Symbol">(</a><a id="76" href="Categories.Category.Core.html#442" class="Record">Category</a><a id="84" class="Symbol">;</a> <a id="86" class="Keyword">module</a> <a id="93" href="Categories.Category.html#1298" class="Module">Commutation</a><a id="104" class="Symbol">)</a>
<a id="106" class="Keyword">open</a> <a id="111" class="Keyword">import</a> <a id="118" href="Categories.Category.Monoidal.html" class="Module">Categories.Category.Monoidal</a>
<a id="147" class="Keyword">open</a> <a id="152" class="Keyword">import</a> <a id="159" href="Categories.Category.Monoidal.Braided.html" class="Module">Categories.Category.Monoidal.Braided</a> <a id="196" class="Keyword">using</a> <a id="202" class="Symbol">(</a><a id="203" href="Categories.Category.Monoidal.Braided.html#643" class="Record">Braided</a><a id="210" class="Symbol">)</a>
<a id="213" class="Keyword">module</a> <a id="220" href="Categories.Category.Monoidal.Braided.Properties.html" class="Module">Categories.Category.Monoidal.Braided.Properties</a>
<a id="270" class="Symbol">{</a><a id="271" href="Categories.Category.Monoidal.Braided.Properties.html#271" class="Bound">o</a> <a id="273" href="Categories.Category.Monoidal.Braided.Properties.html#273" class="Bound">ℓ</a> <a id="275" href="Categories.Category.Monoidal.Braided.Properties.html#275" class="Bound">e</a><a id="276" class="Symbol">}</a> <a id="278" class="Symbol">{</a><a id="279" href="Categories.Category.Monoidal.Braided.Properties.html#279" class="Bound">C</a> <a id="281" class="Symbol">:</a> <a id="283" href="Categories.Category.Core.html#442" class="Record">Category</a> <a id="292" href="Categories.Category.Monoidal.Braided.Properties.html#271" class="Bound">o</a> <a id="294" href="Categories.Category.Monoidal.Braided.Properties.html#273" class="Bound">ℓ</a> <a id="296" href="Categories.Category.Monoidal.Braided.Properties.html#275" class="Bound">e</a><a id="297" class="Symbol">}</a> <a id="299" class="Symbol">{</a><a id="300" href="Categories.Category.Monoidal.Braided.Properties.html#300" class="Bound">M</a> <a id="302" class="Symbol">:</a> <a id="304" href="Categories.Category.Monoidal.Core.html#1287" class="Record">Monoidal</a> <a id="313" href="Categories.Category.Monoidal.Braided.Properties.html#279" class="Bound">C</a><a id="314" class="Symbol">}</a> <a id="316" class="Symbol">(</a><a id="317" href="Categories.Category.Monoidal.Braided.Properties.html#317" class="Bound">BM</a> <a id="320" class="Symbol">:</a> <a id="322" href="Categories.Category.Monoidal.Braided.html#643" class="Record">Braided</a> <a id="330" href="Categories.Category.Monoidal.Braided.Properties.html#300" class="Bound">M</a><a id="331" class="Symbol">)</a> <a id="333" class="Keyword">where</a>
<a id="340" class="Keyword">open</a> <a id="345" class="Keyword">import</a> <a id="352" href="Data.Product.html" class="Module">Data.Product</a> <a id="365" class="Keyword">using</a> <a id="371" class="Symbol">(</a><a id="372" href="Agda.Builtin.Sigma.html#235" class="InductiveConstructor Operator">_,_</a><a id="375" class="Symbol">)</a>
<a id="378" class="Keyword">import</a> <a id="385" href="Categories.Category.Construction.Core.html" class="Module">Categories.Category.Construction.Core</a> <a id="423" href="Categories.Category.Monoidal.Braided.Properties.html#279" class="Bound">C</a> as <a id="Core"></a><a id="428" href="Categories.Category.Monoidal.Braided.Properties.html#428" class="Module">Core</a>
<a id="433" class="Keyword">open</a> <a id="438" class="Keyword">import</a> <a id="445" href="Categories.Category.Monoidal.Properties.html" class="Module">Categories.Category.Monoidal.Properties</a> <a id="485" href="Categories.Category.Monoidal.Braided.Properties.html#300" class="Bound">M</a>
<a id="487" class="Keyword">open</a> <a id="492" class="Keyword">import</a> <a id="499" href="Categories.Category.Monoidal.Reasoning.html" class="Module">Categories.Category.Monoidal.Reasoning</a> <a id="538" href="Categories.Category.Monoidal.Braided.Properties.html#300" class="Bound">M</a>
<a id="540" class="Keyword">import</a> <a id="547" href="Categories.Category.Monoidal.Utilities.html" class="Module">Categories.Category.Monoidal.Utilities</a> <a id="586" href="Categories.Category.Monoidal.Braided.Properties.html#300" class="Bound">M</a> as <a id="MonoidalUtilities"></a><a id="591" href="Categories.Category.Monoidal.Braided.Properties.html#591" class="Module">MonoidalUtilities</a>
<a id="609" class="Keyword">open</a> <a id="614" class="Keyword">import</a> <a id="621" href="Categories.Functor.html" class="Module">Categories.Functor</a> <a id="640" class="Keyword">using</a> <a id="646" class="Symbol">(</a><a id="647" href="Categories.Functor.Core.html#248" class="Record">Functor</a><a id="654" class="Symbol">)</a>
<a id="656" class="Keyword">open</a> <a id="661" class="Keyword">import</a> <a id="668" href="Categories.Morphism.Reasoning.html" class="Module">Categories.Morphism.Reasoning</a> <a id="698" href="Categories.Category.Monoidal.Braided.Properties.html#279" class="Bound">C</a> <a id="700" class="Keyword">hiding</a> <a id="707" class="Symbol">(</a><a id="708" href="Categories.Morphism.Reasoning.Iso.html#4018" class="Function">push-eq</a><a id="715" class="Symbol">)</a>
<a id="717" class="Keyword">open</a> <a id="722" class="Keyword">import</a> <a id="729" href="Categories.NaturalTransformation.NaturalIsomorphism.html" class="Module">Categories.NaturalTransformation.NaturalIsomorphism</a> <a id="781" class="Keyword">using</a> <a id="787" class="Symbol">(</a><a id="788" href="Categories.NaturalTransformation.NaturalIsomorphism.html#2810" class="Function">niHelper</a><a id="796" class="Symbol">)</a>
<a id="798" class="Keyword">open</a> <a id="803" class="Keyword">import</a> <a id="810" href="Categories.NaturalTransformation.NaturalIsomorphism.Properties.html" class="Module">Categories.NaturalTransformation.NaturalIsomorphism.Properties</a>
<a id="875" class="Keyword">using</a> <a id="881" class="Symbol">(</a><a id="882" href="Categories.NaturalTransformation.NaturalIsomorphism.Properties.html#1621" class="Function">push-eq</a><a id="889" class="Symbol">)</a>
<a id="892" class="Keyword">open</a> <a id="897" href="Categories.Category.Core.html#442" class="Module">Category</a> <a id="906" href="Categories.Category.Monoidal.Braided.Properties.html#279" class="Bound">C</a>
<a id="908" class="Keyword">open</a> <a id="913" href="Categories.Category.html#1298" class="Module">Commutation</a> <a id="925" href="Categories.Category.Monoidal.Braided.Properties.html#279" class="Bound">C</a>
<a id="927" class="Keyword">open</a> <a id="932" href="Categories.Category.Monoidal.Braided.html#643" class="Module">Braided</a> <a id="940" href="Categories.Category.Monoidal.Braided.Properties.html#317" class="Bound">BM</a>
<a id="943" class="Keyword">open</a> <a id="948" href="Categories.Category.Monoidal.Braided.Properties.html#591" class="Module">MonoidalUtilities</a> <a id="966" class="Keyword">using</a> <a id="972" class="Symbol">(</a><a id="973" href="Categories.Category.Monoidal.Utilities.html#3689" class="Function Operator">_⊗ᵢ_</a><a id="977" class="Symbol">;</a> <a id="979" href="Categories.Category.Monoidal.Utilities.html#2653" class="Function">unitorʳ-naturalIsomorphism</a><a id="1005" class="Symbol">)</a>
<a id="1007" class="Keyword">open</a> <a id="1012" href="Categories.Category.Monoidal.Utilities.html#1438" class="Module">MonoidalUtilities.Shorthands</a>
<a id="1041" class="Keyword">open</a> <a id="1046" href="Categories.Category.Construction.Core.html#1314" class="Module">Core.Shorthands</a>
<a id="1062" class="Keyword">open</a> <a id="1067" href="Categories.Category.Construction.Core.html#1340" class="Module">Commutationᵢ</a>
<a id="1081" class="Keyword">private</a>
<a id="1091" class="Keyword">variable</a>
<a id="1104" href="Categories.Category.Monoidal.Braided.Properties.html#1104" class="Generalizable">X</a> <a id="1106" href="Categories.Category.Monoidal.Braided.Properties.html#1106" class="Generalizable">Y</a> <a id="1108" href="Categories.Category.Monoidal.Braided.Properties.html#1108" class="Generalizable">Z</a> <a id="1110" class="Symbol">:</a> <a id="1112" href="Categories.Category.Core.html#559" class="Function">Obj</a>
<a id="1117" class="Comment">-- Shorthands for the braiding</a>
<a id="1149" class="Keyword">module</a> <a id="Shorthands"></a><a id="1156" href="Categories.Category.Monoidal.Braided.Properties.html#1156" class="Module">Shorthands</a> <a id="1167" class="Keyword">where</a>
<a id="Shorthands.σ⇒"></a><a id="1176" href="Categories.Category.Monoidal.Braided.Properties.html#1176" class="Function">σ⇒</a> <a id="1179" class="Symbol">:</a> <a id="1181" class="Symbol">∀</a> <a id="1183" class="Symbol">{</a><a id="1184" href="Categories.Category.Monoidal.Braided.Properties.html#1184" class="Bound">X</a> <a id="1186" href="Categories.Category.Monoidal.Braided.Properties.html#1186" class="Bound">Y</a><a id="1187" class="Symbol">}</a> <a id="1189" class="Symbol">→</a> <a id="1191" href="Categories.Category.Monoidal.Braided.Properties.html#1184" class="Bound">X</a> <a id="1193" href="Categories.Category.Monoidal.Core.html#1435" class="Function Operator">⊗₀</a> <a id="1196" href="Categories.Category.Monoidal.Braided.Properties.html#1186" class="Bound">Y</a> <a id="1198" href="Categories.Category.Core.html#575" class="Function Operator">⇒</a> <a id="1200" href="Categories.Category.Monoidal.Braided.Properties.html#1186" class="Bound">Y</a> <a id="1202" href="Categories.Category.Monoidal.Core.html#1435" class="Function Operator">⊗₀</a> <a id="1205" href="Categories.Category.Monoidal.Braided.Properties.html#1184" class="Bound">X</a>
<a id="1209" href="Categories.Category.Monoidal.Braided.Properties.html#1176" class="Function">σ⇒</a> <a id="1212" class="Symbol">{</a><a id="1213" href="Categories.Category.Monoidal.Braided.Properties.html#1213" class="Bound">X</a><a id="1214" class="Symbol">}</a> <a id="1216" class="Symbol">{</a><a id="1217" href="Categories.Category.Monoidal.Braided.Properties.html#1217" class="Bound">Y</a><a id="1218" class="Symbol">}</a> <a id="1220" class="Symbol">=</a> <a id="1222" href="Categories.NaturalTransformation.Core.html#783" class="Function">braiding.⇒.η</a> <a id="1235" class="Symbol">(</a><a id="1236" href="Categories.Category.Monoidal.Braided.Properties.html#1213" class="Bound">X</a> <a id="1238" href="Agda.Builtin.Sigma.html#235" class="InductiveConstructor Operator">,</a> <a id="1240" href="Categories.Category.Monoidal.Braided.Properties.html#1217" class="Bound">Y</a><a id="1241" class="Symbol">)</a>
<a id="Shorthands.σ⇐"></a><a id="1246" href="Categories.Category.Monoidal.Braided.Properties.html#1246" class="Function">σ⇐</a> <a id="1249" class="Symbol">:</a> <a id="1251" class="Symbol">∀</a> <a id="1253" class="Symbol">{</a><a id="1254" href="Categories.Category.Monoidal.Braided.Properties.html#1254" class="Bound">X</a> <a id="1256" href="Categories.Category.Monoidal.Braided.Properties.html#1256" class="Bound">Y</a><a id="1257" class="Symbol">}</a> <a id="1259" class="Symbol">→</a> <a id="1261" href="Categories.Category.Monoidal.Braided.Properties.html#1256" class="Bound">Y</a> <a id="1263" href="Categories.Category.Monoidal.Core.html#1435" class="Function Operator">⊗₀</a> <a id="1266" href="Categories.Category.Monoidal.Braided.Properties.html#1254" class="Bound">X</a> <a id="1268" href="Categories.Category.Core.html#575" class="Function Operator">⇒</a> <a id="1270" href="Categories.Category.Monoidal.Braided.Properties.html#1254" class="Bound">X</a> <a id="1272" href="Categories.Category.Monoidal.Core.html#1435" class="Function Operator">⊗₀</a> <a id="1275" href="Categories.Category.Monoidal.Braided.Properties.html#1256" class="Bound">Y</a>
<a id="1279" href="Categories.Category.Monoidal.Braided.Properties.html#1246" class="Function">σ⇐</a> <a id="1282" class="Symbol">{</a><a id="1283" href="Categories.Category.Monoidal.Braided.Properties.html#1283" class="Bound">X</a><a id="1284" class="Symbol">}</a> <a id="1286" class="Symbol">{</a><a id="1287" href="Categories.Category.Monoidal.Braided.Properties.html#1287" class="Bound">Y</a><a id="1288" class="Symbol">}</a> <a id="1290" class="Symbol">=</a> <a id="1292" href="Categories.NaturalTransformation.Core.html#783" class="Function">braiding.⇐.η</a> <a id="1305" class="Symbol">(</a><a id="1306" href="Categories.Category.Monoidal.Braided.Properties.html#1283" class="Bound">X</a> <a id="1308" href="Agda.Builtin.Sigma.html#235" class="InductiveConstructor Operator">,</a> <a id="1310" href="Categories.Category.Monoidal.Braided.Properties.html#1287" class="Bound">Y</a><a id="1311" class="Symbol">)</a>
<a id="Shorthands.σ"></a><a id="1316" href="Categories.Category.Monoidal.Braided.Properties.html#1316" class="Function">σ</a> <a id="1318" class="Symbol">=</a> <a id="1320" href="Categories.NaturalTransformation.NaturalIsomorphism.html#1155" class="Function">braiding.FX≅GX</a>
<a id="1336" class="Keyword">open</a> <a id="1341" href="Categories.Category.Monoidal.Braided.Properties.html#1156" class="Module">Shorthands</a>
<a id="1353" class="Keyword">private</a>
<a id="1364" class="Comment">-- It&#39;s easier to prove the following lemma, which is the desired</a>
<a id="1432" class="Comment">-- coherence theorem moduolo application of the |-⊗ unit| functor.</a>
<a id="1501" class="Comment">-- Because |-⊗ unit| is equivalent to the identity functor, the</a>
<a id="1567" class="Comment">-- lemma and the theorem are equivalent.</a>
<a id="1611" class="Comment">-- The following diagram illustrates the hexagon that we are</a>
<a id="1674" class="Comment">-- operating on. The main outer hexagon is hexagon₁, the braiding</a>
<a id="1742" class="Comment">-- coherence, instantiated with X, 1 and 1 (Here we denote the unit</a>
<a id="1812" class="Comment">-- by 1 for brevity).</a>
<a id="1836" class="Comment">-- In the middle are X1 and 1X along with morphisms towards them.</a>
<a id="1904" class="Comment">-- The lower hexagon (given by the double lines) commutes and is</a>
<a id="1971" class="Comment">-- an intermediary in the final proof. It is there to effectively</a>
<a id="2039" class="Comment">-- get rid of the top half of the main hexagon.</a>
<a id="2089" class="Comment">-- The rest of the proof is isolating the bottom left triangle</a>
<a id="2154" class="Comment">-- which represents our desired identity. It is doing that by</a>
<a id="2218" class="Comment">-- proving that the pentagon to the right of it commutes.</a>
<a id="2278" class="Comment">-- The pentagon commuting is, in turn, proved by gluing the</a>
<a id="2340" class="Comment">-- rightmost &quot;square&quot; onto the middle triangle.</a>
<a id="2390" class="Comment">--</a>
<a id="2395" class="Comment">--</a>
<a id="2400" class="Comment">-- ┌─────&gt; X(11) ─────────&gt; (11)X ──────┐</a>
<a id="2453" class="Comment">-- ┌┘ α │ σ │ α └┐</a>
<a id="2507" class="Comment">-- ┌┘ │id⊗λ │λ⊗id └┐</a>
<a id="2561" class="Comment">-- ┌┘ V V V</a>
<a id="2616" class="Comment">-- (X1)1 ═══════&gt; X1 ════════════&gt; 1X &lt;══════ 1(1X)</a>
<a id="2673" class="Comment">-- ╚╗ ρ⊗id Λ &lt;───┐ σ λ Λ</a>
<a id="2728" class="Comment">-- ╚╗ │λ⊗id └────────┐ ╔╝</a>
<a id="2783" class="Comment">-- ╚╗ │ λ └┐ ╔╝</a>
<a id="2837" class="Comment">-- ╚═════&gt; (1X)1 ═════════&gt; 1(X1) ═════╝</a>
<a id="2890" class="Comment">-- σ⊗id α id⊗σ</a>
<a id="braiding-coherence⊗unit"></a><a id="2944" href="Categories.Category.Monoidal.Braided.Properties.html#2944" class="Function">braiding-coherence⊗unit</a> <a id="2968" class="Symbol">:</a> <a id="2970" href="Categories.Category.html#1379" class="Function Operator">[</a> <a id="2972" class="Symbol">(</a><a id="2973" href="Categories.Category.Monoidal.Braided.Properties.html#1104" class="Generalizable">X</a> <a id="2975" href="Categories.Category.Monoidal.Core.html#1435" class="Function Operator">⊗₀</a> <a id="2978" href="Categories.Category.Monoidal.Core.html#1380" class="Function">unit</a><a id="2982" class="Symbol">)</a> <a id="2984" href="Categories.Category.Monoidal.Core.html#1435" class="Function Operator">⊗₀</a> <a id="2987" href="Categories.Category.Monoidal.Core.html#1380" class="Function">unit</a> <a id="2992" href="Categories.Category.html#1379" class="Function Operator">⇒</a> <a id="2994" href="Categories.Category.Monoidal.Braided.Properties.html#1104" class="Generalizable">X</a> <a id="2996" href="Categories.Category.Monoidal.Core.html#1435" class="Function Operator">⊗₀</a> <a id="2999" href="Categories.Category.Monoidal.Core.html#1380" class="Function">unit</a> <a id="3004" href="Categories.Category.html#1379" class="Function Operator">]⟨</a>
<a id="3037" href="Categories.Category.Monoidal.Braided.Properties.html#1176" class="Function">σ⇒</a> <a id="3040" href="Categories.Category.Monoidal.Core.html#1536" class="Function Operator">⊗₁</a> <a id="3043" href="Categories.Category.Core.html#630" class="Function">id</a> <a id="3057" href="Categories.Category.html#1481" class="Function">⇒⟨</a> <a id="3060" class="Symbol">(</a><a id="3061" href="Categories.Category.Monoidal.Core.html#1380" class="Function">unit</a> <a id="3066" href="Categories.Category.Monoidal.Core.html#1435" class="Function Operator">⊗₀</a> <a id="3069" href="Categories.Category.Monoidal.Braided.Properties.html#1104" class="Generalizable">X</a><a id="3070" class="Symbol">)</a> <a id="3072" href="Categories.Category.Monoidal.Core.html#1435" class="Function Operator">⊗₀</a> <a id="3075" href="Categories.Category.Monoidal.Core.html#1380" class="Function">unit</a> <a id="3080" href="Categories.Category.html#1481" class="Function">⟩</a>
<a id="3112" href="Categories.Category.Monoidal.Utilities.html#1458" class="Function">λ⇒</a> <a id="3115" href="Categories.Category.Monoidal.Core.html#1536" class="Function Operator">⊗₁</a> <a id="3118" href="Categories.Category.Core.html#630" class="Function">id</a>
<a id="3149" href="Categories.Category.html#1379" class="Function Operator">≈</a> <a id="3151" href="Categories.Category.Monoidal.Utilities.html#1496" class="Function">ρ⇒</a> <a id="3154" href="Categories.Category.Monoidal.Core.html#1536" class="Function Operator">⊗₁</a> <a id="3157" href="Categories.Category.Core.html#630" class="Function">id</a>
<a id="3188" href="Categories.Category.html#1379" class="Function Operator">⟩</a>
<a id="3192" href="Categories.Category.Monoidal.Braided.Properties.html#2944" class="Function">braiding-coherence⊗unit</a> <a id="3216" class="Symbol">=</a> <a id="3218" href="Categories.Morphism.Reasoning.Iso.html#1691" class="Function">cancel-fromˡ</a> <a id="3231" href="Categories.NaturalTransformation.NaturalIsomorphism.html#1155" class="Function">braiding.FX≅GX</a> <a id="3246" class="Symbol">(</a><a id="3247" href="Relation.Binary.Reasoning.Syntax.html#1510" class="Function Operator">begin</a>
<a id="3257" href="Categories.Category.Monoidal.Braided.Properties.html#1176" class="Function">σ⇒</a> <a id="3260" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="3262" href="Categories.Category.Monoidal.Utilities.html#1458" class="Function">λ⇒</a> <a id="3265" href="Categories.Category.Monoidal.Core.html#1536" class="Function Operator">⊗₁</a> <a id="3268" href="Categories.Category.Core.html#630" class="Function">id</a> <a id="3271" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="3273" href="Categories.Category.Monoidal.Braided.Properties.html#1176" class="Function">σ⇒</a> <a id="3276" href="Categories.Category.Monoidal.Core.html#1536" class="Function Operator">⊗₁</a> <a id="3279" href="Categories.Category.Core.html#630" class="Function">id</a> <a id="3293" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="3296" href="Categories.Morphism.Reasoning.Core.html#2048" class="Function">pullˡ</a> <a id="3302" class="Symbol">(</a><a id="3303" href="Categories.Category.Core.html#3005" class="Function">⟺</a> <a id="3305" class="Symbol">(</a><a id="3306" href="Categories.Morphism.Reasoning.Core.html#5531" class="Function">glue◽◃</a> <a id="3313" href="Categories.Category.Monoidal.Core.html#2517" class="Function">unitorˡ-commute-from</a> <a id="3334" href="Categories.Category.Monoidal.Properties.html#4950" class="Function">coherence₁</a><a id="3344" class="Symbol">))</a> <a id="3347" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
<a id="3353" class="Symbol">(</a><a id="3354" href="Categories.Category.Monoidal.Utilities.html#1458" class="Function">λ⇒</a> <a id="3357" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="3359" href="Categories.Category.Core.html#630" class="Function">id</a> <a id="3362" href="Categories.Category.Monoidal.Core.html#1536" class="Function Operator">⊗₁</a> <a id="3365" href="Categories.Category.Monoidal.Braided.Properties.html#1176" class="Function">σ⇒</a> <a id="3368" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="3370" href="Categories.Category.Monoidal.Utilities.html#1596" class="Function">α⇒</a><a id="3372" class="Symbol">)</a> <a id="3374" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="3376" href="Categories.Category.Monoidal.Braided.Properties.html#1176" class="Function">σ⇒</a> <a id="3379" href="Categories.Category.Monoidal.Core.html#1536" class="Function Operator">⊗₁</a> <a id="3382" href="Categories.Category.Core.html#630" class="Function">id</a> <a id="3389" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="3392" href="Categories.Morphism.Reasoning.Core.html#1551" class="Function">assoc²&#39;</a> <a id="3400" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
<a id="3406" href="Categories.Category.Monoidal.Utilities.html#1458" class="Function">λ⇒</a> <a id="3409" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="3411" href="Categories.Category.Core.html#630" class="Function">id</a> <a id="3414" href="Categories.Category.Monoidal.Core.html#1536" class="Function Operator">⊗₁</a> <a id="3417" href="Categories.Category.Monoidal.Braided.Properties.html#1176" class="Function">σ⇒</a> <a id="3420" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="3422" href="Categories.Category.Monoidal.Utilities.html#1596" class="Function">α⇒</a> <a id="3425" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="3427" href="Categories.Category.Monoidal.Braided.Properties.html#1176" class="Function">σ⇒</a> <a id="3430" href="Categories.Category.Monoidal.Core.html#1536" class="Function Operator">⊗₁</a> <a id="3433" href="Categories.Category.Core.html#630" class="Function">id</a> <a id="3442" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="3445" href="Categories.Category.Core.html#2734" class="Function Operator">refl⟩∘⟨</a> <a id="3453" href="Categories.Category.Monoidal.Braided.html#912" class="Field">hexagon₁</a> <a id="3462" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
<a id="3468" href="Categories.Category.Monoidal.Utilities.html#1458" class="Function">λ⇒</a> <a id="3471" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="3473" href="Categories.Category.Monoidal.Utilities.html#1596" class="Function">α⇒</a> <a id="3476" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="3478" href="Categories.Category.Monoidal.Braided.Properties.html#1176" class="Function">σ⇒</a> <a id="3481" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="3483" href="Categories.Category.Monoidal.Utilities.html#1596" class="Function">α⇒</a> <a id="3504" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="3507" href="Categories.Morphism.Reasoning.Core.html#2048" class="Function">pullˡ</a> <a id="3513" href="Categories.Category.Monoidal.Properties.html#4950" class="Function">coherence₁</a> <a id="3524" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
<a id="3530" href="Categories.Category.Monoidal.Utilities.html#1458" class="Function">λ⇒</a> <a id="3533" href="Categories.Category.Monoidal.Core.html#1536" class="Function Operator">⊗₁</a> <a id="3536" href="Categories.Category.Core.html#630" class="Function">id</a> <a id="3539" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="3541" href="Categories.Category.Monoidal.Braided.Properties.html#1176" class="Function">σ⇒</a> <a id="3544" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="3546" href="Categories.Category.Monoidal.Utilities.html#1596" class="Function">α⇒</a> <a id="3566" href="Relation.Binary.Reasoning.Syntax.html#7400" class="Function">≈˘⟨</a> <a id="3570" href="Categories.Morphism.Reasoning.Core.html#2485" class="Function">pushˡ</a> <a id="3576" class="Symbol">(</a><a id="3577" href="Categories.NaturalTransformation.Core.html#827" class="Function">braiding.⇒.commute</a> <a id="3596" class="Symbol">_)</a> <a id="3599" href="Relation.Binary.Reasoning.Syntax.html#7400" class="Function">⟩</a>
<a id="3605" class="Symbol">(</a><a id="3606" href="Categories.Category.Monoidal.Braided.Properties.html#1176" class="Function">σ⇒</a> <a id="3609" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="3611" href="Categories.Category.Core.html#630" class="Function">id</a> <a id="3614" href="Categories.Category.Monoidal.Core.html#1536" class="Function Operator">⊗₁</a> <a id="3617" href="Categories.Category.Monoidal.Utilities.html#1458" class="Function">λ⇒</a><a id="3619" class="Symbol">)</a> <a id="3621" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="3623" href="Categories.Category.Monoidal.Utilities.html#1596" class="Function">α⇒</a> <a id="3641" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="3644" href="Categories.Morphism.Reasoning.Core.html#1914" class="Function">pullʳ</a> <a id="3650" href="Categories.Category.Monoidal.Core.html#2943" class="Function">triangle</a> <a id="3659" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
<a id="3665" href="Categories.Category.Monoidal.Braided.Properties.html#1176" class="Function">σ⇒</a> <a id="3668" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="3670" href="Categories.Category.Monoidal.Utilities.html#1496" class="Function">ρ⇒</a> <a id="3673" href="Categories.Category.Monoidal.Core.html#1536" class="Function Operator">⊗₁</a> <a id="3676" href="Categories.Category.Core.html#630" class="Function">id</a> <a id="3701" href="Relation.Binary.Reasoning.Syntax.html#12283" class="Function Operator">∎</a><a id="3702" class="Symbol">)</a>
<a id="3705" class="Comment">-- The desired theorem follows from |braiding-coherence⊗unit| by</a>
<a id="3770" class="Comment">-- translating it along the right unitor (which is a natural iso).</a>
<a id="braiding-coherence"></a><a id="3838" href="Categories.Category.Monoidal.Braided.Properties.html#3838" class="Function">braiding-coherence</a> <a id="3857" class="Symbol">:</a> <a id="3859" href="Categories.Category.html#1379" class="Function Operator">[</a> <a id="3861" href="Categories.Category.Monoidal.Braided.Properties.html#1104" class="Generalizable">X</a> <a id="3863" href="Categories.Category.Monoidal.Core.html#1435" class="Function Operator">⊗₀</a> <a id="3866" href="Categories.Category.Monoidal.Core.html#1380" class="Function">unit</a> <a id="3871" href="Categories.Category.html#1379" class="Function Operator">⇒</a> <a id="3873" href="Categories.Category.Monoidal.Braided.Properties.html#1104" class="Generalizable">X</a> <a id="3875" href="Categories.Category.html#1379" class="Function Operator">]⟨</a>
<a id="3901" href="Categories.Category.Monoidal.Braided.Properties.html#1176" class="Function">σ⇒</a> <a id="3917" href="Categories.Category.html#1481" class="Function">⇒⟨</a> <a id="3920" href="Categories.Category.Monoidal.Core.html#1380" class="Function">unit</a> <a id="3925" href="Categories.Category.Monoidal.Core.html#1435" class="Function Operator">⊗₀</a> <a id="3928" href="Categories.Category.Monoidal.Braided.Properties.html#1104" class="Generalizable">X</a> <a id="3930" href="Categories.Category.html#1481" class="Function">⟩</a>
<a id="3955" href="Categories.Category.Monoidal.Utilities.html#1458" class="Function">λ⇒</a>
<a id="3979" href="Categories.Category.html#1379" class="Function Operator">≈</a> <a id="3981" href="Categories.Category.Monoidal.Utilities.html#1496" class="Function">ρ⇒</a>
<a id="4005" href="Categories.Category.html#1379" class="Function Operator">⟩</a>
<a id="4007" href="Categories.Category.Monoidal.Braided.Properties.html#3838" class="Function">braiding-coherence</a> <a id="4026" class="Symbol">=</a> <a id="4028" href="Categories.NaturalTransformation.NaturalIsomorphism.Properties.html#1621" class="Function">push-eq</a> <a id="4036" href="Categories.Category.Monoidal.Utilities.html#2653" class="Function">unitorʳ-naturalIsomorphism</a> <a id="4063" class="Symbol">(</a><a id="4064" href="Relation.Binary.Reasoning.Syntax.html#1510" class="Function Operator">begin</a>
<a id="4072" class="Symbol">(</a><a id="4073" href="Categories.Category.Monoidal.Utilities.html#1458" class="Function">λ⇒</a> <a id="4076" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="4078" href="Categories.Category.Monoidal.Braided.Properties.html#1176" class="Function">σ⇒</a><a id="4080" class="Symbol">)</a> <a id="4082" href="Categories.Category.Monoidal.Core.html#1536" class="Function Operator">⊗₁</a> <a id="4085" href="Categories.Category.Core.html#630" class="Function">id</a> <a id="4098" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="4101" href="Categories.Functor.Core.html#565" class="Function">homomorphism</a> <a id="4114" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
<a id="4118" class="Symbol">(</a><a id="4119" href="Categories.Category.Monoidal.Utilities.html#1458" class="Function">λ⇒</a> <a id="4122" href="Categories.Category.Monoidal.Core.html#1536" class="Function Operator">⊗₁</a> <a id="4125" href="Categories.Category.Core.html#630" class="Function">id</a><a id="4127" class="Symbol">)</a> <a id="4129" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="4131" class="Symbol">(</a><a id="4132" href="Categories.Category.Monoidal.Braided.Properties.html#1176" class="Function">σ⇒</a> <a id="4135" href="Categories.Category.Monoidal.Core.html#1536" class="Function Operator">⊗₁</a> <a id="4138" href="Categories.Category.Core.html#630" class="Function">id</a><a id="4140" class="Symbol">)</a> <a id="4144" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="4147" href="Categories.Category.Monoidal.Braided.Properties.html#2944" class="Function">braiding-coherence⊗unit</a> <a id="4171" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
<a id="4175" href="Categories.Category.Monoidal.Utilities.html#1496" class="Function">ρ⇒</a> <a id="4179" href="Categories.Category.Monoidal.Core.html#1536" class="Function Operator">⊗₁</a> <a id="4182" href="Categories.Category.Core.html#630" class="Function">id</a> <a id="4201" href="Relation.Binary.Reasoning.Syntax.html#12283" class="Function Operator">∎</a><a id="4202" class="Symbol">)</a>
<a id="4206" class="Keyword">where</a> <a id="4212" class="Keyword">open</a> <a id="4217" href="Categories.Functor.Core.html#248" class="Module">Functor</a> <a id="4225" class="Symbol">(</a><a id="4226" href="Categories.Category.Monoidal.Core.html#1645" class="Function Operator">-⊗</a> <a id="4229" href="Categories.Category.Monoidal.Core.html#1380" class="Function">unit</a><a id="4233" class="Symbol">)</a>
<a id="4236" class="Comment">-- Variants of the hexagon identities defined on isos.</a>
<a id="hexagon₁-iso"></a><a id="4292" href="Categories.Category.Monoidal.Braided.Properties.html#4292" class="Function">hexagon₁-iso</a> <a id="4305" class="Symbol">:</a> <a id="4307" href="Categories.Category.Core.html#630" class="Function">idᵢ</a> <a id="4311" href="Categories.Category.Monoidal.Utilities.html#3689" class="Function Operator">⊗ᵢ</a> <a id="4314" href="Categories.Category.Monoidal.Braided.Properties.html#1316" class="Function">σ</a> <a id="4316" href="Categories.Category.Core.html#656" class="Function Operator">∘ᵢ</a> <a id="4319" href="Categories.Category.Monoidal.Core.html#1762" class="Function">associator</a> <a id="4330" href="Categories.Category.Core.html#656" class="Function Operator">∘ᵢ</a> <a id="4333" href="Categories.Category.Monoidal.Braided.Properties.html#1316" class="Function">σ</a> <a id="4335" class="Symbol">{</a><a id="4336" href="Categories.Category.Monoidal.Braided.Properties.html#1104" class="Generalizable">X</a> <a id="4338" href="Agda.Builtin.Sigma.html#235" class="InductiveConstructor Operator">,</a> <a id="4340" href="Categories.Category.Monoidal.Braided.Properties.html#1106" class="Generalizable">Y</a><a id="4341" class="Symbol">}</a> <a id="4343" href="Categories.Category.Monoidal.Utilities.html#3689" class="Function Operator">⊗ᵢ</a> <a id="4346" href="Categories.Category.Core.html#630" class="Function">idᵢ</a> <a id="4350" class="Symbol">{</a><a id="4351" href="Categories.Category.Monoidal.Braided.Properties.html#1108" class="Generalizable">Z</a><a id="4352" class="Symbol">}</a> <a id="4354" href="Categories.Category.Core.html#595" class="Function Operator">≈ᵢ</a>
<a id="4372" href="Categories.Category.Monoidal.Core.html#1762" class="Function">associator</a> <a id="4383" href="Categories.Category.Core.html#656" class="Function Operator">∘ᵢ</a> <a id="4386" href="Categories.Category.Monoidal.Braided.Properties.html#1316" class="Function">σ</a> <a id="4388" class="Symbol">{</a><a id="4389" href="Categories.Category.Monoidal.Braided.Properties.html#1104" class="Generalizable">X</a> <a id="4391" href="Agda.Builtin.Sigma.html#235" class="InductiveConstructor Operator">,</a> <a id="4393" href="Categories.Category.Monoidal.Braided.Properties.html#1106" class="Generalizable">Y</a> <a id="4395" href="Categories.Category.Monoidal.Core.html#1435" class="Function Operator">⊗₀</a> <a id="4398" href="Categories.Category.Monoidal.Braided.Properties.html#1108" class="Generalizable">Z</a><a id="4399" class="Symbol">}</a> <a id="4401" href="Categories.Category.Core.html#656" class="Function Operator">∘ᵢ</a> <a id="4404" href="Categories.Category.Monoidal.Core.html#1762" class="Function">associator</a>
<a id="4415" href="Categories.Category.Monoidal.Braided.Properties.html#4292" class="Function">hexagon₁-iso</a> <a id="4428" class="Symbol">=</a> <a id="4430" href="Categories.Morphism.IsoEquiv.html#1807" class="InductiveConstructor Operator">⌞</a> <a id="4432" href="Categories.Category.Monoidal.Braided.html#912" class="Field">hexagon₁</a> <a id="4441" href="Categories.Morphism.IsoEquiv.html#1807" class="InductiveConstructor Operator">⌟</a>
<a id="hexagon₁-inv"></a><a id="4444" href="Categories.Category.Monoidal.Braided.Properties.html#4444" class="Function">hexagon₁-inv</a> <a id="4457" class="Symbol">:</a> <a id="4459" class="Symbol">(</a><a id="4460" href="Categories.Category.Monoidal.Braided.Properties.html#1246" class="Function">σ⇐</a> <a id="4463" class="Symbol">{</a><a id="4464" href="Categories.Category.Monoidal.Braided.Properties.html#1104" class="Generalizable">X</a><a id="4465" class="Symbol">}</a> <a id="4467" class="Symbol">{</a><a id="4468" href="Categories.Category.Monoidal.Braided.Properties.html#1106" class="Generalizable">Y</a><a id="4469" class="Symbol">}</a> <a id="4471" href="Categories.Category.Monoidal.Core.html#1536" class="Function Operator">⊗₁</a> <a id="4474" href="Categories.Category.Core.html#630" class="Function">id</a> <a id="4477" class="Symbol">{</a><a id="4478" href="Categories.Category.Monoidal.Braided.Properties.html#1108" class="Generalizable">Z</a><a id="4479" class="Symbol">}</a> <a id="4481" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="4483" href="Categories.Category.Monoidal.Utilities.html#1647" class="Function">α⇐</a><a id="4485" class="Symbol">)</a> <a id="4487" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="4489" href="Categories.Category.Core.html#630" class="Function">id</a> <a id="4492" href="Categories.Category.Monoidal.Core.html#1536" class="Function Operator">⊗₁</a> <a id="4495" href="Categories.Category.Monoidal.Braided.Properties.html#1246" class="Function">σ⇐</a> <a id="4498" href="Categories.Category.Core.html#595" class="Function Operator">≈</a>
<a id="4515" class="Symbol">(</a><a id="4516" href="Categories.Category.Monoidal.Utilities.html#1647" class="Function">α⇐</a> <a id="4519" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="4521" href="Categories.Category.Monoidal.Braided.Properties.html#1246" class="Function">σ⇐</a> <a id="4524" class="Symbol">{</a><a id="4525" href="Categories.Category.Monoidal.Braided.Properties.html#1104" class="Generalizable">X</a><a id="4526" class="Symbol">}</a> <a id="4528" class="Symbol">{</a><a id="4529" href="Categories.Category.Monoidal.Braided.Properties.html#1106" class="Generalizable">Y</a> <a id="4531" href="Categories.Category.Monoidal.Core.html#1435" class="Function Operator">⊗₀</a> <a id="4534" href="Categories.Category.Monoidal.Braided.Properties.html#1108" class="Generalizable">Z</a><a id="4535" class="Symbol">})</a> <a id="4538" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="4540" href="Categories.Category.Monoidal.Utilities.html#1647" class="Function">α⇐</a>
<a id="4543" href="Categories.Category.Monoidal.Braided.Properties.html#4444" class="Function">hexagon₁-inv</a> <a id="4556" class="Symbol">=</a> <a id="4558" href="Categories.Morphism.IsoEquiv.html#1858" class="Function">to-≈</a> <a id="4563" href="Categories.Category.Monoidal.Braided.Properties.html#4292" class="Function">hexagon₁-iso</a>
<a id="hexagon₂-iso"></a><a id="4577" href="Categories.Category.Monoidal.Braided.Properties.html#4577" class="Function">hexagon₂-iso</a> <a id="4590" class="Symbol">:</a> <a id="4592" class="Symbol">(</a><a id="4593" href="Categories.Category.Monoidal.Braided.Properties.html#1316" class="Function">σ</a> <a id="4595" href="Categories.Category.Monoidal.Utilities.html#3689" class="Function Operator">⊗ᵢ</a> <a id="4598" href="Categories.Category.Core.html#630" class="Function">idᵢ</a> <a id="4602" href="Categories.Category.Core.html#656" class="Function Operator">∘ᵢ</a> <a id="4605" href="Categories.Category.Monoidal.Core.html#1762" class="Function">associator</a> <a id="4616" href="Categories.Category.Groupoid.html#365" class="Function Operator">⁻¹</a><a id="4618" class="Symbol">)</a> <a id="4620" href="Categories.Category.Core.html#656" class="Function Operator">∘ᵢ</a> <a id="4623" href="Categories.Category.Core.html#630" class="Function">idᵢ</a> <a id="4627" class="Symbol">{</a><a id="4628" href="Categories.Category.Monoidal.Braided.Properties.html#1104" class="Generalizable">X</a><a id="4629" class="Symbol">}</a> <a id="4631" href="Categories.Category.Monoidal.Utilities.html#3689" class="Function Operator">⊗ᵢ</a> <a id="4634" href="Categories.Category.Monoidal.Braided.Properties.html#1316" class="Function">σ</a> <a id="4636" class="Symbol">{</a><a id="4637" href="Categories.Category.Monoidal.Braided.Properties.html#1106" class="Generalizable">Y</a> <a id="4639" href="Agda.Builtin.Sigma.html#235" class="InductiveConstructor Operator">,</a> <a id="4641" href="Categories.Category.Monoidal.Braided.Properties.html#1108" class="Generalizable">Z</a><a id="4642" class="Symbol">}</a> <a id="4644" href="Categories.Category.Core.html#595" class="Function Operator">≈ᵢ</a>
<a id="4662" class="Symbol">(</a><a id="4663" href="Categories.Category.Monoidal.Core.html#1762" class="Function">associator</a> <a id="4674" href="Categories.Category.Groupoid.html#365" class="Function Operator">⁻¹</a> <a id="4677" href="Categories.Category.Core.html#656" class="Function Operator">∘ᵢ</a> <a id="4680" href="Categories.Category.Monoidal.Braided.Properties.html#1316" class="Function">σ</a> <a id="4682" class="Symbol">{</a><a id="4683" href="Categories.Category.Monoidal.Braided.Properties.html#1104" class="Generalizable">X</a> <a id="4685" href="Categories.Category.Monoidal.Core.html#1435" class="Function Operator">⊗₀</a> <a id="4688" href="Categories.Category.Monoidal.Braided.Properties.html#1106" class="Generalizable">Y</a> <a id="4690" href="Agda.Builtin.Sigma.html#235" class="InductiveConstructor Operator">,</a> <a id="4692" href="Categories.Category.Monoidal.Braided.Properties.html#1108" class="Generalizable">Z</a><a id="4693" class="Symbol">})</a> <a id="4696" href="Categories.Category.Core.html#656" class="Function Operator">∘ᵢ</a> <a id="4699" href="Categories.Category.Monoidal.Core.html#1762" class="Function">associator</a> <a id="4710" href="Categories.Category.Groupoid.html#365" class="Function Operator">⁻¹</a>
<a id="4713" href="Categories.Category.Monoidal.Braided.Properties.html#4577" class="Function">hexagon₂-iso</a> <a id="4726" class="Symbol">=</a> <a id="4728" href="Categories.Morphism.IsoEquiv.html#1807" class="InductiveConstructor Operator">⌞</a> <a id="4730" href="Categories.Category.Monoidal.Braided.html#1287" class="Field">hexagon₂</a> <a id="4739" href="Categories.Morphism.IsoEquiv.html#1807" class="InductiveConstructor Operator">⌟</a>
<a id="hexagon₂-inv"></a><a id="4742" href="Categories.Category.Monoidal.Braided.Properties.html#4742" class="Function">hexagon₂-inv</a> <a id="4755" class="Symbol">:</a> <a id="4757" href="Categories.Category.Core.html#630" class="Function">id</a> <a id="4760" class="Symbol">{</a><a id="4761" href="Categories.Category.Monoidal.Braided.Properties.html#1104" class="Generalizable">X</a><a id="4762" class="Symbol">}</a> <a id="4764" href="Categories.Category.Monoidal.Core.html#1536" class="Function Operator">⊗₁</a> <a id="4767" href="Categories.Category.Monoidal.Braided.Properties.html#1246" class="Function">σ⇐</a> <a id="4770" class="Symbol">{</a><a id="4771" href="Categories.Category.Monoidal.Braided.Properties.html#1106" class="Generalizable">Y</a><a id="4772" class="Symbol">}</a> <a id="4774" class="Symbol">{</a><a id="4775" href="Categories.Category.Monoidal.Braided.Properties.html#1108" class="Generalizable">Z</a><a id="4776" class="Symbol">}</a> <a id="4778" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="4780" href="Categories.Category.Monoidal.Utilities.html#1596" class="Function">α⇒</a> <a id="4783" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="4785" href="Categories.Category.Monoidal.Braided.Properties.html#1246" class="Function">σ⇐</a> <a id="4788" href="Categories.Category.Monoidal.Core.html#1536" class="Function Operator">⊗₁</a> <a id="4791" href="Categories.Category.Core.html#630" class="Function">id</a> <a id="4794" href="Categories.Category.Core.html#595" class="Function Operator">≈</a>
<a id="4811" href="Categories.Category.Monoidal.Utilities.html#1596" class="Function">α⇒</a> <a id="4814" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="4816" href="Categories.Category.Monoidal.Braided.Properties.html#1246" class="Function">σ⇐</a> <a id="4819" class="Symbol">{</a><a id="4820" href="Categories.Category.Monoidal.Braided.Properties.html#1104" class="Generalizable">X</a> <a id="4822" href="Categories.Category.Monoidal.Core.html#1435" class="Function Operator">⊗₀</a> <a id="4825" href="Categories.Category.Monoidal.Braided.Properties.html#1106" class="Generalizable">Y</a><a id="4826" class="Symbol">}</a> <a id="4828" class="Symbol">{</a><a id="4829" href="Categories.Category.Monoidal.Braided.Properties.html#1108" class="Generalizable">Z</a><a id="4830" class="Symbol">}</a> <a id="4832" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="4834" href="Categories.Category.Monoidal.Utilities.html#1596" class="Function">α⇒</a>
<a id="4837" href="Categories.Category.Monoidal.Braided.Properties.html#4742" class="Function">hexagon₂-inv</a> <a id="4850" class="Symbol">=</a> <a id="4852" href="Categories.Morphism.IsoEquiv.html#1858" class="Function">to-≈</a> <a id="4857" href="Categories.Category.Monoidal.Braided.Properties.html#4577" class="Function">hexagon₂-iso</a>
<a id="4871" class="Comment">-- Variants of the above coherence law.</a>
<a id="braiding-coherence-iso"></a><a id="4912" href="Categories.Category.Monoidal.Braided.Properties.html#4912" class="Function">braiding-coherence-iso</a> <a id="4935" class="Symbol">:</a> <a id="4937" href="Categories.Category.Monoidal.Core.html#1700" class="Function">unitorˡ</a> <a id="4945" href="Categories.Category.Core.html#656" class="Function Operator">∘ᵢ</a> <a id="4948" href="Categories.Category.Monoidal.Braided.Properties.html#1316" class="Function">σ</a> <a id="4950" href="Categories.Category.Core.html#595" class="Function Operator">≈ᵢ</a> <a id="4953" href="Categories.Category.Monoidal.Core.html#1731" class="Function">unitorʳ</a> <a id="4961" class="Symbol">{</a><a id="4962" href="Categories.Category.Monoidal.Braided.Properties.html#1104" class="Generalizable">X</a><a id="4963" class="Symbol">}</a>
<a id="4965" href="Categories.Category.Monoidal.Braided.Properties.html#4912" class="Function">braiding-coherence-iso</a> <a id="4988" class="Symbol">=</a> <a id="4990" href="Categories.Morphism.IsoEquiv.html#1807" class="InductiveConstructor Operator">⌞</a> <a id="4992" href="Categories.Category.Monoidal.Braided.Properties.html#3838" class="Function">braiding-coherence</a> <a id="5011" href="Categories.Morphism.IsoEquiv.html#1807" class="InductiveConstructor Operator">⌟</a>
<a id="braiding-coherence-inv"></a><a id="5014" href="Categories.Category.Monoidal.Braided.Properties.html#5014" class="Function">braiding-coherence-inv</a> <a id="5037" class="Symbol">:</a> <a id="5039" href="Categories.Category.Monoidal.Braided.Properties.html#1246" class="Function">σ⇐</a> <a id="5042" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="5044" href="Categories.Category.Monoidal.Utilities.html#1478" class="Function">λ⇐</a> <a id="5047" href="Categories.Category.Core.html#595" class="Function Operator">≈</a> <a id="5049" href="Categories.Category.Monoidal.Utilities.html#1516" class="Function">ρ⇐</a> <a id="5052" class="Symbol">{</a><a id="5053" href="Categories.Category.Monoidal.Braided.Properties.html#1104" class="Generalizable">X</a><a id="5054" class="Symbol">}</a>
<a id="5056" href="Categories.Category.Monoidal.Braided.Properties.html#5014" class="Function">braiding-coherence-inv</a> <a id="5079" class="Symbol">=</a> <a id="5081" href="Categories.Morphism.IsoEquiv.html#1858" class="Function">to-≈</a> <a id="5086" href="Categories.Category.Monoidal.Braided.Properties.html#4912" class="Function">braiding-coherence-iso</a>
<a id="5110" class="Comment">-- The inverse of the braiding is also a braiding on M.</a>
<a id="inv-Braided"></a><a id="5167" href="Categories.Category.Monoidal.Braided.Properties.html#5167" class="Function">inv-Braided</a> <a id="5179" class="Symbol">:</a> <a id="5181" href="Categories.Category.Monoidal.Braided.html#643" class="Record">Braided</a> <a id="5189" href="Categories.Category.Monoidal.Braided.Properties.html#300" class="Bound">M</a>
<a id="5191" href="Categories.Category.Monoidal.Braided.Properties.html#5167" class="Function">inv-Braided</a> <a id="5203" class="Symbol">=</a> <a id="5205" class="Keyword">record</a>
<a id="5214" class="Symbol">{</a> <a id="5216" href="Categories.Category.Monoidal.Braided.html#717" class="Field">braiding</a> <a id="5225" class="Symbol">=</a> <a id="5227" href="Categories.NaturalTransformation.NaturalIsomorphism.html#2810" class="Function">niHelper</a> <a id="5236" class="Symbol">(</a><a id="5237" class="Keyword">record</a>
<a id="5248" class="Symbol">{</a> <a id="5250" href="Categories.NaturalTransformation.NaturalIsomorphism.html#2606" class="Field">η</a> <a id="5258" class="Symbol">=</a> <a id="5260" class="Symbol">λ</a> <a id="5262" href="Categories.Category.Monoidal.Braided.Properties.html#5262" class="Bound">_</a> <a id="5264" class="Symbol">→</a> <a id="5266" href="Categories.Category.Monoidal.Braided.Properties.html#1246" class="Function">σ⇐</a>
<a id="5273" class="Symbol">;</a> <a id="5275" href="Categories.NaturalTransformation.NaturalIsomorphism.html#2648" class="Field">η⁻¹</a> <a id="5283" class="Symbol">=</a> <a id="5285" class="Symbol">λ</a> <a id="5287" href="Categories.Category.Monoidal.Braided.Properties.html#5287" class="Bound">_</a> <a id="5289" class="Symbol">→</a> <a id="5291" href="Categories.Category.Monoidal.Braided.Properties.html#1176" class="Function">σ⇒</a>
<a id="5298" class="Symbol">;</a> <a id="5300" href="Categories.NaturalTransformation.NaturalIsomorphism.html#2690" class="Field">commute</a> <a id="5308" class="Symbol">=</a> <a id="5310" class="Symbol">λ{</a> <a id="5313" class="Symbol">(</a><a id="5314" href="Categories.Category.Monoidal.Braided.Properties.html#5314" class="Bound">f</a> <a id="5316" href="Agda.Builtin.Sigma.html#235" class="InductiveConstructor Operator">,</a> <a id="5318" href="Categories.Category.Monoidal.Braided.Properties.html#5318" class="Bound">g</a><a id="5319" class="Symbol">)</a> <a id="5321" class="Symbol">→</a> <a id="5323" href="Categories.NaturalTransformation.Core.html#827" class="Function">braiding.⇐.commute</a> <a id="5342" class="Symbol">(</a><a id="5343" href="Categories.Category.Monoidal.Braided.Properties.html#5318" class="Bound">g</a> <a id="5345" href="Agda.Builtin.Sigma.html#235" class="InductiveConstructor Operator">,</a> <a id="5347" href="Categories.Category.Monoidal.Braided.Properties.html#5314" class="Bound">f</a><a id="5348" class="Symbol">)</a> <a id="5350" class="Symbol">}</a>
<a id="5356" class="Symbol">;</a> <a id="5358" href="Categories.NaturalTransformation.NaturalIsomorphism.html#2760" class="Field">iso</a> <a id="5366" class="Symbol">=</a> <a id="5368" class="Symbol">λ{</a> <a id="5371" class="Symbol">(</a><a id="5372" href="Categories.Category.Monoidal.Braided.Properties.html#5372" class="Bound">X</a> <a id="5374" href="Agda.Builtin.Sigma.html#235" class="InductiveConstructor Operator">,</a> <a id="5376" href="Categories.Category.Monoidal.Braided.Properties.html#5376" class="Bound">Y</a><a id="5377" class="Symbol">)</a> <a id="5379" class="Symbol">→</a> <a id="5381" class="Keyword">record</a>
<a id="5394" class="Symbol">{</a> <a id="5396" href="Categories.Morphism.html#1586" class="Field">isoˡ</a> <a id="5401" class="Symbol">=</a> <a id="5403" href="Categories.Morphism.html#1612" class="Function">braiding.iso.isoʳ</a> <a id="5421" class="Symbol">(</a><a id="5422" href="Categories.Category.Monoidal.Braided.Properties.html#5376" class="Bound">Y</a> <a id="5424" href="Agda.Builtin.Sigma.html#235" class="InductiveConstructor Operator">,</a> <a id="5426" href="Categories.Category.Monoidal.Braided.Properties.html#5372" class="Bound">X</a><a id="5427" class="Symbol">)</a>
<a id="5435" class="Symbol">;</a> <a id="5437" href="Categories.Morphism.html#1612" class="Field">isoʳ</a> <a id="5442" class="Symbol">=</a> <a id="5444" href="Categories.Morphism.html#1586" class="Function">braiding.iso.isoˡ</a> <a id="5462" class="Symbol">(</a><a id="5463" href="Categories.Category.Monoidal.Braided.Properties.html#5376" class="Bound">Y</a> <a id="5465" href="Agda.Builtin.Sigma.html#235" class="InductiveConstructor Operator">,</a> <a id="5467" href="Categories.Category.Monoidal.Braided.Properties.html#5372" class="Bound">X</a><a id="5468" class="Symbol">)</a> <a id="5470" class="Symbol">}</a> <a id="5472" class="Symbol">}</a>
<a id="5478" class="Symbol">})</a>
<a id="5483" class="Symbol">;</a> <a id="5485" href="Categories.Category.Monoidal.Braided.html#912" class="Field">hexagon₁</a> <a id="5494" class="Symbol">=</a> <a id="5496" href="Categories.Category.Monoidal.Braided.Properties.html#4742" class="Function">hexagon₂-inv</a>
<a id="5511" class="Symbol">;</a> <a id="5513" href="Categories.Category.Monoidal.Braided.html#1287" class="Field">hexagon₂</a> <a id="5522" class="Symbol">=</a> <a id="5524" href="Categories.Category.Monoidal.Braided.Properties.html#4444" class="Function">hexagon₁-inv</a>
<a id="5539" class="Symbol">}</a>
<a id="5542" class="Comment">-- A variant of the above coherence law for the inverse of the braiding.</a>
<a id="inv-braiding-coherence"></a><a id="5616" href="Categories.Category.Monoidal.Braided.Properties.html#5616" class="Function">inv-braiding-coherence</a> <a id="5639" class="Symbol">:</a> <a id="5641" href="Categories.Category.html#1379" class="Function Operator">[</a> <a id="5643" href="Categories.Category.Monoidal.Core.html#1380" class="Function">unit</a> <a id="5648" href="Categories.Category.Monoidal.Core.html#1435" class="Function Operator">⊗₀</a> <a id="5651" href="Categories.Category.Monoidal.Braided.Properties.html#1104" class="Generalizable">X</a> <a id="5653" href="Categories.Category.html#1379" class="Function Operator">⇒</a> <a id="5655" href="Categories.Category.Monoidal.Braided.Properties.html#1104" class="Generalizable">X</a> <a id="5657" href="Categories.Category.html#1379" class="Function Operator">]⟨</a>
<a id="5687" href="Categories.Category.Monoidal.Braided.Properties.html#1246" class="Function">σ⇐</a> <a id="5701" href="Categories.Category.html#1481" class="Function">⇒⟨</a> <a id="5704" href="Categories.Category.Monoidal.Braided.Properties.html#1104" class="Generalizable">X</a> <a id="5706" href="Categories.Category.Monoidal.Core.html#1435" class="Function Operator">⊗₀</a> <a id="5709" href="Categories.Category.Monoidal.Core.html#1380" class="Function">unit</a> <a id="5714" href="Categories.Category.html#1481" class="Function">⟩</a>
<a id="5743" href="Categories.Category.Monoidal.Utilities.html#1496" class="Function">ρ⇒</a>
<a id="5771" href="Categories.Category.html#1379" class="Function Operator">≈</a> <a id="5773" href="Categories.Category.Monoidal.Utilities.html#1458" class="Function">λ⇒</a>
<a id="5801" href="Categories.Category.html#1379" class="Function Operator">⟩</a>
<a id="5803" href="Categories.Category.Monoidal.Braided.Properties.html#5616" class="Function">inv-braiding-coherence</a> <a id="5826" class="Symbol">=</a> <a id="5828" href="Categories.Category.Core.html#3005" class="Function">⟺</a> <a id="5830" class="Symbol">(</a><a id="5831" href="Categories.Morphism.Reasoning.Iso.html#1094" class="Function">switch-fromtoʳ</a> <a id="5846" href="Categories.Category.Monoidal.Braided.Properties.html#1316" class="Function">σ</a> <a id="5848" href="Categories.Category.Monoidal.Braided.Properties.html#3838" class="Function">braiding-coherence</a><a id="5866" class="Symbol">)</a>
<a id="5869" class="Comment">-- Reversing a ternary product via braiding commutes with the associator.</a>
<a id="assoc-reverse"></a><a id="5944" href="Categories.Category.Monoidal.Braided.Properties.html#5944" class="Function">assoc-reverse</a> <a id="5958" class="Symbol">:</a> <a id="5960" href="Categories.Category.html#1379" class="Function Operator">[</a> <a id="5962" href="Categories.Category.Monoidal.Braided.Properties.html#1104" class="Generalizable">X</a> <a id="5964" href="Categories.Category.Monoidal.Core.html#1435" class="Function Operator">⊗₀</a> <a id="5967" class="Symbol">(</a><a id="5968" href="Categories.Category.Monoidal.Braided.Properties.html#1106" class="Generalizable">Y</a> <a id="5970" href="Categories.Category.Monoidal.Core.html#1435" class="Function Operator">⊗₀</a> <a id="5973" href="Categories.Category.Monoidal.Braided.Properties.html#1108" class="Generalizable">Z</a><a id="5974" class="Symbol">)</a> <a id="5976" href="Categories.Category.html#1379" class="Function Operator">⇒</a> <a id="5978" class="Symbol">(</a><a id="5979" href="Categories.Category.Monoidal.Braided.Properties.html#1104" class="Generalizable">X</a> <a id="5981" href="Categories.Category.Monoidal.Core.html#1435" class="Function Operator">⊗₀</a> <a id="5984" href="Categories.Category.Monoidal.Braided.Properties.html#1106" class="Generalizable">Y</a><a id="5985" class="Symbol">)</a> <a id="5987" href="Categories.Category.Monoidal.Core.html#1435" class="Function Operator">⊗₀</a> <a id="5990" href="Categories.Category.Monoidal.Braided.Properties.html#1108" class="Generalizable">Z</a> <a id="5992" href="Categories.Category.html#1379" class="Function Operator">]⟨</a>
<a id="6013" href="Categories.Category.Core.html#630" class="Function">id</a> <a id="6016" href="Categories.Category.Monoidal.Core.html#1536" class="Function Operator">⊗₁</a> <a id="6019" href="Categories.Category.Monoidal.Braided.Properties.html#1176" class="Function">σ⇒</a> <a id="6027" href="Categories.Category.html#1481" class="Function">⇒⟨</a> <a id="6030" href="Categories.Category.Monoidal.Braided.Properties.html#1104" class="Generalizable">X</a> <a id="6032" href="Categories.Category.Monoidal.Core.html#1435" class="Function Operator">⊗₀</a> <a id="6035" class="Symbol">(</a><a id="6036" href="Categories.Category.Monoidal.Braided.Properties.html#1108" class="Generalizable">Z</a> <a id="6038" href="Categories.Category.Monoidal.Core.html#1435" class="Function Operator">⊗₀</a> <a id="6041" href="Categories.Category.Monoidal.Braided.Properties.html#1106" class="Generalizable">Y</a><a id="6042" class="Symbol">)</a> <a id="6044" href="Categories.Category.html#1481" class="Function">⟩</a>
<a id="6064" href="Categories.Category.Monoidal.Braided.Properties.html#1176" class="Function">σ⇒</a> <a id="6078" href="Categories.Category.html#1481" class="Function">⇒⟨</a> <a id="6081" class="Symbol">(</a><a id="6082" href="Categories.Category.Monoidal.Braided.Properties.html#1108" class="Generalizable">Z</a> <a id="6084" href="Categories.Category.Monoidal.Core.html#1435" class="Function Operator">⊗₀</a> <a id="6087" href="Categories.Category.Monoidal.Braided.Properties.html#1106" class="Generalizable">Y</a><a id="6088" class="Symbol">)</a> <a id="6090" href="Categories.Category.Monoidal.Core.html#1435" class="Function Operator">⊗₀</a> <a id="6093" href="Categories.Category.Monoidal.Braided.Properties.html#1104" class="Generalizable">X</a> <a id="6095" href="Categories.Category.html#1481" class="Function">⟩</a>
<a id="6115" href="Categories.Category.Monoidal.Utilities.html#1596" class="Function">α⇒</a> <a id="6129" href="Categories.Category.html#1481" class="Function">⇒⟨</a> <a id="6132" href="Categories.Category.Monoidal.Braided.Properties.html#1108" class="Generalizable">Z</a> <a id="6134" href="Categories.Category.Monoidal.Core.html#1435" class="Function Operator">⊗₀</a> <a id="6137" class="Symbol">(</a><a id="6138" href="Categories.Category.Monoidal.Braided.Properties.html#1106" class="Generalizable">Y</a> <a id="6140" href="Categories.Category.Monoidal.Core.html#1435" class="Function Operator">⊗₀</a> <a id="6143" href="Categories.Category.Monoidal.Braided.Properties.html#1104" class="Generalizable">X</a><a id="6144" class="Symbol">)</a> <a id="6146" href="Categories.Category.html#1481" class="Function">⟩</a>
<a id="6166" href="Categories.Category.Core.html#630" class="Function">id</a> <a id="6169" href="Categories.Category.Monoidal.Core.html#1536" class="Function Operator">⊗₁</a> <a id="6172" href="Categories.Category.Monoidal.Braided.Properties.html#1246" class="Function">σ⇐</a> <a id="6180" href="Categories.Category.html#1481" class="Function">⇒⟨</a> <a id="6183" href="Categories.Category.Monoidal.Braided.Properties.html#1108" class="Generalizable">Z</a> <a id="6185" href="Categories.Category.Monoidal.Core.html#1435" class="Function Operator">⊗₀</a> <a id="6188" class="Symbol">(</a><a id="6189" href="Categories.Category.Monoidal.Braided.Properties.html#1104" class="Generalizable">X</a> <a id="6191" href="Categories.Category.Monoidal.Core.html#1435" class="Function Operator">⊗₀</a> <a id="6194" href="Categories.Category.Monoidal.Braided.Properties.html#1106" class="Generalizable">Y</a><a id="6195" class="Symbol">)</a> <a id="6197" href="Categories.Category.html#1481" class="Function">⟩</a>
<a id="6217" href="Categories.Category.Monoidal.Braided.Properties.html#1246" class="Function">σ⇐</a>
<a id="6236" href="Categories.Category.html#1379" class="Function Operator">≈</a> <a id="6238" href="Categories.Category.Monoidal.Utilities.html#1647" class="Function">α⇐</a>
<a id="6257" href="Categories.Category.html#1379" class="Function Operator">⟩</a>
<a id="6259" href="Categories.Category.Monoidal.Braided.Properties.html#5944" class="Function">assoc-reverse</a> <a id="6273" class="Symbol">=</a> <a id="6275" href="Relation.Binary.Reasoning.Syntax.html#1510" class="Function Operator">begin</a>
<a id="6283" href="Categories.Category.Monoidal.Braided.Properties.html#1246" class="Function">σ⇐</a> <a id="6286" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="6288" href="Categories.Category.Core.html#630" class="Function">id</a> <a id="6291" href="Categories.Category.Monoidal.Core.html#1536" class="Function Operator">⊗₁</a> <a id="6294" href="Categories.Category.Monoidal.Braided.Properties.html#1246" class="Function">σ⇐</a> <a id="6297" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="6299" href="Categories.Category.Monoidal.Utilities.html#1596" class="Function">α⇒</a> <a id="6302" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="6304" href="Categories.Category.Monoidal.Braided.Properties.html#1176" class="Function">σ⇒</a> <a id="6307" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="6309" href="Categories.Category.Core.html#630" class="Function">id</a> <a id="6312" href="Categories.Category.Monoidal.Core.html#1536" class="Function Operator">⊗₁</a> <a id="6315" href="Categories.Category.Monoidal.Braided.Properties.html#1176" class="Function">σ⇒</a> <a id="6321" href="Relation.Binary.Reasoning.Syntax.html#7400" class="Function">≈˘⟨</a> <a id="6325" href="Categories.Category.Core.html#2734" class="Function Operator">refl⟩∘⟨</a> <a id="6333" href="Categories.Morphism.Reasoning.Core.html#1551" class="Function">assoc²&#39;</a> <a id="6341" href="Relation.Binary.Reasoning.Syntax.html#7400" class="Function">⟩</a>
<a id="6345" href="Categories.Category.Monoidal.Braided.Properties.html#1246" class="Function">σ⇐</a> <a id="6348" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="6350" class="Symbol">(</a><a id="6351" href="Categories.Category.Core.html#630" class="Function">id</a> <a id="6354" href="Categories.Category.Monoidal.Core.html#1536" class="Function Operator">⊗₁</a> <a id="6357" href="Categories.Category.Monoidal.Braided.Properties.html#1246" class="Function">σ⇐</a> <a id="6360" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="6362" href="Categories.Category.Monoidal.Utilities.html#1596" class="Function">α⇒</a> <a id="6365" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="6367" href="Categories.Category.Monoidal.Braided.Properties.html#1176" class="Function">σ⇒</a><a id="6369" class="Symbol">)</a> <a id="6371" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="6373" href="Categories.Category.Core.html#630" class="Function">id</a> <a id="6376" href="Categories.Category.Monoidal.Core.html#1536" class="Function Operator">⊗₁</a> <a id="6379" href="Categories.Category.Monoidal.Braided.Properties.html#1176" class="Function">σ⇒</a> <a id="6383" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="6386" href="Categories.Category.Core.html#2734" class="Function Operator">refl⟩∘⟨</a> <a id="6394" href="Categories.Morphism.Reasoning.Core.html#2485" class="Function">pushˡ</a> <a id="6400" href="Categories.Category.Monoidal.Braided.Properties.html#6704" class="Function">hex₁&#39;</a> <a id="6406" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
<a id="6410" href="Categories.Category.Monoidal.Braided.Properties.html#1246" class="Function">σ⇐</a> <a id="6413" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="6415" class="Symbol">(</a><a id="6416" href="Categories.Category.Monoidal.Utilities.html#1596" class="Function">α⇒</a> <a id="6419" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="6421" href="Categories.Category.Monoidal.Braided.Properties.html#1176" class="Function">σ⇒</a> <a id="6424" href="Categories.Category.Monoidal.Core.html#1536" class="Function Operator">⊗₁</a> <a id="6427" href="Categories.Category.Core.html#630" class="Function">id</a><a id="6429" class="Symbol">)</a> <a id="6431" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="6433" href="Categories.Category.Monoidal.Utilities.html#1647" class="Function">α⇐</a> <a id="6436" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="6438" href="Categories.Category.Core.html#630" class="Function">id</a> <a id="6441" href="Categories.Category.Monoidal.Core.html#1536" class="Function Operator">⊗₁</a> <a id="6444" href="Categories.Category.Monoidal.Braided.Properties.html#1176" class="Function">σ⇒</a> <a id="6448" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="6451" href="Categories.Category.Core.html#2734" class="Function Operator">refl⟩∘⟨</a> <a id="6459" href="Categories.Morphism.Reasoning.Core.html#1914" class="Function">pullʳ</a> <a id="6465" class="Symbol">(</a><a id="6466" href="Categories.Category.Core.html#1004" class="Function">sym-assoc</a> <a id="6476" href="Categories.Category.Core.html#3061" class="Function Operator">○</a> <a id="6478" href="Categories.Category.Monoidal.Braided.html#1287" class="Field">hexagon₂</a><a id="6486" class="Symbol">)</a> <a id="6488" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
<a id="6492" href="Categories.Category.Monoidal.Braided.Properties.html#1246" class="Function">σ⇐</a> <a id="6495" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="6497" href="Categories.Category.Monoidal.Utilities.html#1596" class="Function">α⇒</a> <a id="6500" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="6502" class="Symbol">(</a><a id="6503" href="Categories.Category.Monoidal.Utilities.html#1647" class="Function">α⇐</a> <a id="6506" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="6508" href="Categories.Category.Monoidal.Braided.Properties.html#1176" class="Function">σ⇒</a><a id="6510" class="Symbol">)</a> <a id="6512" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="6514" href="Categories.Category.Monoidal.Utilities.html#1647" class="Function">α⇐</a> <a id="6530" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="6533" href="Categories.Category.Core.html#2734" class="Function Operator">refl⟩∘⟨</a> <a id="6541" href="Categories.Morphism.Reasoning.Core.html#2048" class="Function">pullˡ</a> <a id="6547" class="Symbol">(</a><a id="6548" href="Categories.Morphism.Reasoning.Core.html#6851" class="Function">cancelˡ</a> <a id="6556" href="Categories.Morphism.html#1612" class="Function">associator.isoʳ</a><a id="6571" class="Symbol">)</a> <a id="6573" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
<a id="6577" href="Categories.Category.Monoidal.Braided.Properties.html#1246" class="Function">σ⇐</a> <a id="6580" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="6582" href="Categories.Category.Monoidal.Braided.Properties.html#1176" class="Function">σ⇒</a> <a id="6585" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="6587" href="Categories.Category.Monoidal.Utilities.html#1647" class="Function">α⇐</a> <a id="6615" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="6618" href="Categories.Morphism.Reasoning.Core.html#6851" class="Function">cancelˡ</a> <a id="6626" class="Symbol">(</a><a id="6627" href="Categories.Morphism.html#1586" class="Function">braiding.iso.isoˡ</a> <a id="6645" class="Symbol">_)</a> <a id="6648" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
<a id="6652" href="Categories.Category.Monoidal.Utilities.html#1647" class="Function">α⇐</a> <a id="6690" href="Relation.Binary.Reasoning.Syntax.html#12283" class="Function Operator">∎</a>
<a id="6694" class="Keyword">where</a>
<a id="6704" href="Categories.Category.Monoidal.Braided.Properties.html#6704" class="Function">hex₁&#39;</a> <a id="6710" class="Symbol">=</a> <a id="6712" href="Categories.Morphism.Reasoning.Iso.html#4537" class="Function">conjugate-from</a> <a id="6727" href="Categories.Category.Monoidal.Core.html#1762" class="Function">associator</a> <a id="6738" class="Symbol">(</a><a id="6739" href="Categories.Category.Core.html#630" class="Function">idᵢ</a> <a id="6743" href="Categories.Category.Monoidal.Utilities.html#3689" class="Function Operator">⊗ᵢ</a> <a id="6746" href="Categories.Category.Monoidal.Braided.Properties.html#1316" class="Function">σ</a><a id="6747" class="Symbol">)</a> <a id="6749" class="Symbol">(</a><a id="6750" href="Categories.Category.Core.html#3005" class="Function">⟺</a> <a id="6752" class="Symbol">(</a><a id="6753" href="Categories.Category.Monoidal.Braided.html#912" class="Field">hexagon₁</a> <a id="6762" href="Categories.Category.Core.html#3061" class="Function Operator">○</a> <a id="6764" href="Categories.Category.Core.html#1004" class="Function">sym-assoc</a><a id="6773" class="Symbol">))</a>
</pre></body></html>
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