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<html><head><meta charset="utf-8"><title>Categories.Category.Cartesian.SymmetricMonoidal</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.Cartesian.html" class="Module">Categories.Category.Cartesian</a> <a id="148" class="Keyword">using</a> <a id="154" class="Symbol">(</a><a id="155" href="Categories.Category.Cartesian.html#727" class="Record">Cartesian</a><a id="164" class="Symbol">)</a>
<a id="167" class="Comment">-- Defines the following properties of a Category:</a>
<a id="218" class="Comment">-- Cartesian.SymmetricMonoidal</a>
<a id="249" class="Comment">-- a Cartesian category is Symmetric Monoidal if its induced monoidal structure is symmetric</a>
<a id="346" class="Keyword">module</a> <a id="353" href="Categories.Category.Cartesian.SymmetricMonoidal.html" class="Module">Categories.Category.Cartesian.SymmetricMonoidal</a> <a id="401" class="Symbol">{</a><a id="402" href="Categories.Category.Cartesian.SymmetricMonoidal.html#402" class="Bound">o</a> <a id="404" href="Categories.Category.Cartesian.SymmetricMonoidal.html#404" class="Bound">ℓ</a> <a id="406" href="Categories.Category.Cartesian.SymmetricMonoidal.html#406" class="Bound">e</a><a id="407" class="Symbol">}</a> <a id="409" class="Symbol">(</a><a id="410" href="Categories.Category.Cartesian.SymmetricMonoidal.html#410" class="Bound">𝒞</a> <a id="412" class="Symbol">:</a> <a id="414" href="Categories.Category.Core.html#442" class="Record">Category</a> <a id="423" href="Categories.Category.Cartesian.SymmetricMonoidal.html#402" class="Bound">o</a> <a id="425" href="Categories.Category.Cartesian.SymmetricMonoidal.html#404" class="Bound">ℓ</a> <a id="427" href="Categories.Category.Cartesian.SymmetricMonoidal.html#406" class="Bound">e</a><a id="428" class="Symbol">)</a> <a id="430" class="Symbol">(</a><a id="431" href="Categories.Category.Cartesian.SymmetricMonoidal.html#431" class="Bound">cartesian</a> <a id="441" class="Symbol">:</a> <a id="443" href="Categories.Category.Cartesian.html#727" class="Record">Cartesian</a> <a id="453" href="Categories.Category.Cartesian.SymmetricMonoidal.html#410" class="Bound">𝒞</a><a id="454" class="Symbol">)</a> <a id="456" class="Keyword">where</a>
<a id="463" class="Keyword">open</a> <a id="468" class="Keyword">import</a> <a id="475" href="Data.Product.html" class="Module">Data.Product</a> <a id="488" class="Keyword">using</a> <a id="494" class="Symbol">(</a><a id="495" href="Agda.Builtin.Sigma.html#218" class="InductiveConstructor Operator">_,_</a><a id="498" class="Symbol">)</a>
<a id="501" class="Keyword">open</a> <a id="506" href="Categories.Category.Core.html#442" class="Module">Category</a> <a id="515" href="Categories.Category.Cartesian.SymmetricMonoidal.html#410" class="Bound">𝒞</a>
<a id="517" class="Keyword">open</a> <a id="522" href="Categories.Category.html#1298" class="Module">Commutation</a> <a id="534" href="Categories.Category.Cartesian.SymmetricMonoidal.html#410" class="Bound">𝒞</a>
<a id="536" class="Keyword">open</a> <a id="541" href="Categories.Category.Core.html#2462" class="Module">HomReasoning</a>
<a id="555" class="Keyword">open</a> <a id="560" class="Keyword">import</a> <a id="567" href="Categories.Category.BinaryProducts.html" class="Module">Categories.Category.BinaryProducts</a> <a id="602" href="Categories.Category.Cartesian.SymmetricMonoidal.html#410" class="Bound">𝒞</a> <a id="604" class="Keyword">using</a> <a id="610" class="Symbol">(</a><a id="611" class="Keyword">module</a> <a id="618" href="Categories.Category.BinaryProducts.html#812" class="Module">BinaryProducts</a><a id="632" class="Symbol">)</a>
<a id="634" class="Keyword">open</a> <a id="639" class="Keyword">import</a> <a id="646" href="Categories.Category.Cartesian.Monoidal.html" class="Module">Categories.Category.Cartesian.Monoidal</a> <a id="685" class="Keyword">using</a> <a id="691" class="Symbol">(</a><a id="692" class="Keyword">module</a> <a id="699" href="Categories.Category.Cartesian.Monoidal.html#1139" class="Module">CartesianMonoidal</a><a id="716" class="Symbol">)</a>
<a id="718" class="Keyword">open</a> <a id="723" class="Keyword">import</a> <a id="730" href="Categories.Category.Monoidal.html" class="Module">Categories.Category.Monoidal</a> <a id="759" class="Keyword">using</a> <a id="765" class="Symbol">(</a><a id="766" href="Categories.Category.Monoidal.Core.html#1287" class="Record">Monoidal</a><a id="774" class="Symbol">)</a>
<a id="776" class="Keyword">import</a> <a id="783" href="Categories.Category.Monoidal.Symmetric.html" class="Module">Categories.Category.Monoidal.Symmetric</a> <a id="822" class="Symbol">as</a> <a id="825" class="Module">Sym</a>
<a id="830" class="Keyword">open</a> <a id="835" class="Keyword">import</a> <a id="842" href="Categories.NaturalTransformation.html" class="Module">Categories.NaturalTransformation</a> <a id="875" class="Keyword">using</a> <a id="881" class="Symbol">(</a><a id="882" href="Categories.NaturalTransformation.Core.html#1750" class="Function">ntHelper</a><a id="890" class="Symbol">)</a>
<a id="893" class="Keyword">private</a>
<a id="903" class="Keyword">variable</a>
<a id="916" href="Categories.Category.Cartesian.SymmetricMonoidal.html#916" class="Generalizable">W</a> <a id="918" href="Categories.Category.Cartesian.SymmetricMonoidal.html#918" class="Generalizable">X</a> <a id="920" href="Categories.Category.Cartesian.SymmetricMonoidal.html#920" class="Generalizable">Y</a> <a id="922" href="Categories.Category.Cartesian.SymmetricMonoidal.html#922" class="Generalizable">Z</a> <a id="924" class="Symbol">:</a> <a id="926" href="Categories.Category.Core.html#559" class="Function">Obj</a>
<a id="931" class="Keyword">open</a> <a id="936" href="Categories.Category.Cartesian.html#727" class="Module">Cartesian</a> <a id="946" href="Categories.Category.Cartesian.SymmetricMonoidal.html#431" class="Bound">cartesian</a> <a id="956" class="Keyword">using</a> <a id="962" class="Symbol">(</a><a id="963" href="Categories.Category.Cartesian.html#801" class="Field">products</a><a id="971" class="Symbol">)</a>
<a id="973" class="Keyword">open</a> <a id="978" href="Categories.Category.Cartesian.Monoidal.html#1139" class="Module">CartesianMonoidal</a> <a id="996" href="Categories.Category.Cartesian.SymmetricMonoidal.html#431" class="Bound">cartesian</a> <a id="1006" class="Keyword">using</a> <a id="1012" class="Symbol">(</a><a id="1013" href="Categories.Category.Cartesian.Monoidal.html#5400" class="Function">monoidal</a><a id="1021" class="Symbol">)</a>
<a id="1023" class="Keyword">open</a> <a id="1028" href="Categories.Category.Monoidal.Symmetric.html" class="Module">Sym</a> <a id="1032" href="Categories.Category.Cartesian.Monoidal.html#5400" class="Function">monoidal</a> <a id="1041" class="Keyword">using</a> <a id="1047" class="Symbol">(</a><a id="1048" href="Categories.Category.Monoidal.Symmetric.html#997" class="Record">Symmetric</a><a id="1057" class="Symbol">;</a> <a id="1059" href="Categories.Category.Monoidal.Symmetric.html#3126" class="Function">symmetricHelper</a><a id="1074" class="Symbol">)</a>
<a id="1076" class="Keyword">open</a> <a id="1081" href="Categories.Category.Monoidal.Core.html#1287" class="Module">Monoidal</a> <a id="1090" href="Categories.Category.Cartesian.Monoidal.html#5400" class="Function">monoidal</a> <a id="1099" class="Keyword">using</a> <a id="1105" class="Symbol">(</a><a id="1106" href="Categories.Category.Monoidal.Core.html#1435" class="Function Operator">_⊗₀_</a><a id="1110" class="Symbol">;</a> <a id="1112" href="Categories.Category.Monoidal.Core.html#1536" class="Function Operator">_⊗₁_</a><a id="1116" class="Symbol">;</a> <a id="1118" class="Keyword">module</a> <a id="1125" href="Categories.Category.Monoidal.Core.html#1905" class="Module">associator</a><a id="1135" class="Symbol">)</a>
<a id="1137" class="Keyword">open</a> <a id="1142" href="Categories.Category.BinaryProducts.html#812" class="Module">BinaryProducts</a> <a id="1157" href="Categories.Category.Cartesian.html#801" class="Field">products</a>
<a id="1167" class="Keyword">private</a>
<a id="B"></a><a id="1177" href="Categories.Category.Cartesian.SymmetricMonoidal.html#1177" class="Function">B</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.Cartesian.SymmetricMonoidal.html#1184" class="Bound">X</a> <a id="1186" href="Categories.Category.Cartesian.SymmetricMonoidal.html#1186" class="Bound">Y</a><a id="1187" class="Symbol">}</a> <a id="1189" class="Symbol">→</a> <a id="1191" href="Categories.Category.Cartesian.SymmetricMonoidal.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.Cartesian.SymmetricMonoidal.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.Cartesian.SymmetricMonoidal.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.Cartesian.SymmetricMonoidal.html#1184" class="Bound">X</a>
<a id="1209" href="Categories.Category.Cartesian.SymmetricMonoidal.html#1177" class="Function">B</a> <a id="1211" class="Symbol">=</a> <a id="1213" href="Categories.Category.BinaryProducts.html#2012" class="Function">swap</a>
<a id="hexagon"></a><a id="1219" href="Categories.Category.Cartesian.SymmetricMonoidal.html#1219" class="Function">hexagon</a> <a id="1227" class="Symbol">:</a> <a id="1229" href="Categories.Category.html#1379" class="Function Operator">[</a> <a id="1231" class="Symbol">(</a><a id="1232" href="Categories.Category.Cartesian.SymmetricMonoidal.html#918" class="Generalizable">X</a> <a id="1234" href="Categories.Category.Monoidal.Core.html#1435" class="Function Operator">⊗₀</a> <a id="1237" href="Categories.Category.Cartesian.SymmetricMonoidal.html#920" class="Generalizable">Y</a><a id="1238" class="Symbol">)</a> <a id="1240" href="Categories.Category.Monoidal.Core.html#1435" class="Function Operator">⊗₀</a> <a id="1243" href="Categories.Category.Cartesian.SymmetricMonoidal.html#922" class="Generalizable">Z</a> <a id="1245" href="Categories.Category.html#1379" class="Function Operator">⇒</a> <a id="1247" href="Categories.Category.Cartesian.SymmetricMonoidal.html#920" class="Generalizable">Y</a> <a id="1249" href="Categories.Category.Monoidal.Core.html#1435" class="Function Operator">⊗₀</a> <a id="1252" href="Categories.Category.Cartesian.SymmetricMonoidal.html#922" class="Generalizable">Z</a> <a id="1254" href="Categories.Category.Monoidal.Core.html#1435" class="Function Operator">⊗₀</a> <a id="1257" href="Categories.Category.Cartesian.SymmetricMonoidal.html#918" class="Generalizable">X</a> <a id="1259" href="Categories.Category.html#1379" class="Function Operator">]⟨</a>
<a id="1274" href="Categories.Category.Cartesian.SymmetricMonoidal.html#1177" class="Function">B</a> <a id="1277" href="Categories.Category.Monoidal.Core.html#1536" class="Function Operator">⊗₁</a> <a id="1280" href="Categories.Category.Core.html#630" class="Function">id</a> <a id="1302" href="Categories.Category.html#1481" class="Function">⇒⟨</a> <a id="1305" class="Symbol">(</a><a id="1306" href="Categories.Category.Cartesian.SymmetricMonoidal.html#920" class="Generalizable">Y</a> <a id="1308" href="Categories.Category.Monoidal.Core.html#1435" class="Function Operator">⊗₀</a> <a id="1311" href="Categories.Category.Cartesian.SymmetricMonoidal.html#918" class="Generalizable">X</a><a id="1312" class="Symbol">)</a> <a id="1314" href="Categories.Category.Monoidal.Core.html#1435" class="Function Operator">⊗₀</a> <a id="1317" href="Categories.Category.Cartesian.SymmetricMonoidal.html#922" class="Generalizable">Z</a> <a id="1319" href="Categories.Category.html#1481" class="Function">⟩</a>
<a id="1333" href="Categories.Morphism.html#2006" class="Function">associator.from</a> <a id="1361" href="Categories.Category.html#1481" class="Function">⇒⟨</a> <a id="1364" href="Categories.Category.Cartesian.SymmetricMonoidal.html#920" class="Generalizable">Y</a> <a id="1366" href="Categories.Category.Monoidal.Core.html#1435" class="Function Operator">⊗₀</a> <a id="1369" href="Categories.Category.Cartesian.SymmetricMonoidal.html#918" class="Generalizable">X</a> <a id="1371" href="Categories.Category.Monoidal.Core.html#1435" class="Function Operator">⊗₀</a> <a id="1374" href="Categories.Category.Cartesian.SymmetricMonoidal.html#922" class="Generalizable">Z</a> <a id="1376" href="Categories.Category.html#1481" class="Function">⟩</a>
<a id="1390" href="Categories.Category.Core.html#630" class="Function">id</a> <a id="1393" href="Categories.Category.Monoidal.Core.html#1536" class="Function Operator">⊗₁</a> <a id="1396" href="Categories.Category.Cartesian.SymmetricMonoidal.html#1177" class="Function">B</a>
<a id="1408" href="Categories.Category.html#1379" class="Function Operator">≈</a> <a id="1410" href="Categories.Morphism.html#2006" class="Function">associator.from</a> <a id="1438" href="Categories.Category.html#1481" class="Function">⇒⟨</a> <a id="1441" href="Categories.Category.Cartesian.SymmetricMonoidal.html#918" class="Generalizable">X</a> <a id="1443" href="Categories.Category.Monoidal.Core.html#1435" class="Function Operator">⊗₀</a> <a id="1446" href="Categories.Category.Cartesian.SymmetricMonoidal.html#920" class="Generalizable">Y</a> <a id="1448" href="Categories.Category.Monoidal.Core.html#1435" class="Function Operator">⊗₀</a> <a id="1451" href="Categories.Category.Cartesian.SymmetricMonoidal.html#922" class="Generalizable">Z</a> <a id="1453" href="Categories.Category.html#1481" class="Function">⟩</a>
<a id="1467" href="Categories.Category.Cartesian.SymmetricMonoidal.html#1177" class="Function">B</a> <a id="1495" href="Categories.Category.html#1481" class="Function">⇒⟨</a> <a id="1498" class="Symbol">(</a><a id="1499" href="Categories.Category.Cartesian.SymmetricMonoidal.html#920" class="Generalizable">Y</a> <a id="1501" href="Categories.Category.Monoidal.Core.html#1435" class="Function Operator">⊗₀</a> <a id="1504" href="Categories.Category.Cartesian.SymmetricMonoidal.html#922" class="Generalizable">Z</a><a id="1505" class="Symbol">)</a> <a id="1507" href="Categories.Category.Monoidal.Core.html#1435" class="Function Operator">⊗₀</a> <a id="1510" href="Categories.Category.Cartesian.SymmetricMonoidal.html#918" class="Generalizable">X</a> <a id="1512" href="Categories.Category.html#1481" class="Function">⟩</a>
<a id="1526" href="Categories.Morphism.html#2006" class="Function">associator.from</a>
<a id="1552" href="Categories.Category.html#1379" class="Function Operator">⟩</a>
<a id="1554" href="Categories.Category.Cartesian.SymmetricMonoidal.html#1219" class="Function">hexagon</a> <a id="1562" class="Symbol">=</a> <a id="1564" href="Relation.Binary.Reasoning.Base.Single.html#1925" class="Function Operator">begin</a>
<a id="1576" href="Categories.Category.Core.html#630" class="Function">id</a> <a id="1579" href="Categories.Category.Monoidal.Core.html#1536" class="Function Operator">⊗₁</a> <a id="1582" href="Categories.Category.BinaryProducts.html#2012" class="Function">swap</a> <a id="1587" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="1589" href="Categories.Category.BinaryProducts.html#1506" class="Function">assocˡ</a> <a id="1596" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="1598" href="Categories.Category.BinaryProducts.html#2012" class="Function">swap</a> <a id="1603" href="Categories.Category.Monoidal.Core.html#1536" class="Function Operator">⊗₁</a> <a id="1606" href="Categories.Category.Core.html#630" class="Function">id</a> <a id="1632" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">≈⟨</a> <a id="1635" href="Categories.Category.Core.html#2734" class="Function Operator">refl⟩∘⟨</a> <a id="1643" href="Categories.Category.Core.html#2734" class="Function Operator">refl⟩∘⟨</a> <a id="1651" href="Categories.Category.BinaryProducts.html#1842" class="Function">⟨⟩-congʳ</a> <a id="1660" href="Categories.Category.BinaryProducts.html#2939" class="Function">⟨⟩∘</a> <a id="1664" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">⟩</a>
<a id="1672" href="Categories.Category.Core.html#630" class="Function">id</a> <a id="1675" href="Categories.Category.Monoidal.Core.html#1536" class="Function Operator">⊗₁</a> <a id="1678" href="Categories.Category.BinaryProducts.html#2012" class="Function">swap</a> <a id="1683" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="1685" href="Categories.Category.BinaryProducts.html#1506" class="Function">assocˡ</a> <a id="1692" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="1694" href="Categories.Category.BinaryProducts.html#1360" class="Function Operator">⟨</a> <a id="1696" href="Categories.Category.BinaryProducts.html#1360" class="Function Operator">⟨</a> <a id="1698" href="Categories.Object.Product.Core.html#546" class="Function">π₂</a> <a id="1701" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="1703" href="Categories.Object.Product.Core.html#526" class="Function">π₁</a> <a id="1706" href="Categories.Category.BinaryProducts.html#1360" class="Function Operator">,</a> <a id="1708" href="Categories.Object.Product.Core.html#526" class="Function">π₁</a> <a id="1711" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="1713" href="Categories.Object.Product.Core.html#526" class="Function">π₁</a> <a id="1716" href="Categories.Category.BinaryProducts.html#1360" class="Function Operator">⟩</a> <a id="1718" href="Categories.Category.BinaryProducts.html#1360" class="Function Operator">,</a> <a id="1720" href="Categories.Category.Core.html#630" class="Function">id</a> <a id="1723" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="1725" href="Categories.Object.Product.Core.html#546" class="Function">π₂</a> <a id="1728" href="Categories.Category.BinaryProducts.html#1360" class="Function Operator">⟩</a> <a id="1730" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">≈⟨</a> <a id="1733" href="Categories.Category.Core.html#2734" class="Function Operator">refl⟩∘⟨</a> <a id="1741" href="Categories.Category.BinaryProducts.html#5365" class="Function">assocˡ∘⟨⟩</a> <a id="1751" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">⟩</a>
<a id="1759" href="Categories.Category.Core.html#630" class="Function">id</a> <a id="1762" href="Categories.Category.Monoidal.Core.html#1536" class="Function Operator">⊗₁</a> <a id="1765" href="Categories.Category.BinaryProducts.html#2012" class="Function">swap</a> <a id="1770" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="1772" href="Categories.Category.BinaryProducts.html#1360" class="Function Operator">⟨</a> <a id="1774" href="Categories.Object.Product.Core.html#546" class="Function">π₂</a> <a id="1777" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="1779" href="Categories.Object.Product.Core.html#526" class="Function">π₁</a> <a id="1782" href="Categories.Category.BinaryProducts.html#1360" class="Function Operator">,</a> <a id="1784" href="Categories.Category.BinaryProducts.html#1360" class="Function Operator">⟨</a> <a id="1786" href="Categories.Object.Product.Core.html#526" class="Function">π₁</a> <a id="1789" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="1791" href="Categories.Object.Product.Core.html#526" class="Function">π₁</a> <a id="1794" href="Categories.Category.BinaryProducts.html#1360" class="Function Operator">,</a> <a id="1796" href="Categories.Category.Core.html#630" class="Function">id</a> <a id="1799" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="1801" href="Categories.Object.Product.Core.html#546" class="Function">π₂</a> <a id="1804" href="Categories.Category.BinaryProducts.html#1360" class="Function Operator">⟩</a> <a id="1806" href="Categories.Category.BinaryProducts.html#1360" class="Function Operator">⟩</a> <a id="1817" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">≈⟨</a> <a id="1820" href="Categories.Category.BinaryProducts.html#2563" class="Function">⁂∘⟨⟩</a> <a id="1825" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">⟩</a>
<a id="1833" href="Categories.Category.BinaryProducts.html#1360" class="Function Operator">⟨</a> <a id="1835" href="Categories.Category.Core.html#630" class="Function">id</a> <a id="1838" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="1840" href="Categories.Object.Product.Core.html#546" class="Function">π₂</a> <a id="1843" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="1845" href="Categories.Object.Product.Core.html#526" class="Function">π₁</a> <a id="1848" href="Categories.Category.BinaryProducts.html#1360" class="Function Operator">,</a> <a id="1850" href="Categories.Category.BinaryProducts.html#2012" class="Function">swap</a> <a id="1855" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="1857" href="Categories.Category.BinaryProducts.html#1360" class="Function Operator">⟨</a> <a id="1859" href="Categories.Object.Product.Core.html#526" class="Function">π₁</a> <a id="1862" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="1864" href="Categories.Object.Product.Core.html#526" class="Function">π₁</a> <a id="1867" href="Categories.Category.BinaryProducts.html#1360" class="Function Operator">,</a> <a id="1869" href="Categories.Category.Core.html#630" class="Function">id</a> <a id="1872" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="1874" href="Categories.Object.Product.Core.html#546" class="Function">π₂</a> <a id="1877" href="Categories.Category.BinaryProducts.html#1360" class="Function Operator">⟩</a> <a id="1879" href="Categories.Category.BinaryProducts.html#1360" class="Function Operator">⟩</a> <a id="1891" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">≈⟨</a> <a id="1894" href="Categories.Object.Product.Core.html#854" class="Function">⟨⟩-cong₂</a> <a id="1903" href="Categories.Category.Core.html#1096" class="Function">identityˡ</a> <a id="1913" href="Categories.Category.BinaryProducts.html#4237" class="Function">swap∘⟨⟩</a> <a id="1921" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">⟩</a>
<a id="1929" href="Categories.Category.BinaryProducts.html#1360" class="Function Operator">⟨</a> <a id="1931" href="Categories.Object.Product.Core.html#546" class="Function">π₂</a> <a id="1934" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="1936" href="Categories.Object.Product.Core.html#526" class="Function">π₁</a> <a id="1939" href="Categories.Category.BinaryProducts.html#1360" class="Function Operator">,</a> <a id="1941" href="Categories.Category.BinaryProducts.html#1360" class="Function Operator">⟨</a> <a id="1943" href="Categories.Category.Core.html#630" class="Function">id</a> <a id="1946" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="1948" href="Categories.Object.Product.Core.html#546" class="Function">π₂</a> <a id="1951" href="Categories.Category.BinaryProducts.html#1360" class="Function Operator">,</a> <a id="1953" href="Categories.Object.Product.Core.html#526" class="Function">π₁</a> <a id="1956" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="1958" href="Categories.Object.Product.Core.html#526" class="Function">π₁</a> <a id="1961" href="Categories.Category.BinaryProducts.html#1360" class="Function Operator">⟩</a> <a id="1963" href="Categories.Category.BinaryProducts.html#1360" class="Function Operator">⟩</a> <a id="1987" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">≈⟨</a> <a id="1990" href="Categories.Category.BinaryProducts.html#1927" class="Function">⟨⟩-congˡ</a> <a id="1999" class="Symbol">(</a><a id="2000" href="Categories.Category.BinaryProducts.html#1842" class="Function">⟨⟩-congʳ</a> <a id="2009" href="Categories.Category.Core.html#1096" class="Function">identityˡ</a><a id="2018" class="Symbol">)</a> <a id="2020" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">⟩</a>
<a id="2028" href="Categories.Category.BinaryProducts.html#1360" class="Function Operator">⟨</a> <a id="2030" href="Categories.Object.Product.Core.html#546" class="Function">π₂</a> <a id="2033" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="2035" href="Categories.Object.Product.Core.html#526" class="Function">π₁</a> <a id="2038" href="Categories.Category.BinaryProducts.html#1360" class="Function Operator">,</a> <a id="2040" href="Categories.Category.BinaryProducts.html#1360" class="Function Operator">⟨</a> <a id="2042" href="Categories.Object.Product.Core.html#546" class="Function">π₂</a> <a id="2045" href="Categories.Category.BinaryProducts.html#1360" class="Function Operator">,</a> <a id="2047" href="Categories.Object.Product.Core.html#526" class="Function">π₁</a> <a id="2050" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="2052" href="Categories.Object.Product.Core.html#526" class="Function">π₁</a> <a id="2055" href="Categories.Category.BinaryProducts.html#1360" class="Function Operator">⟩</a> <a id="2057" href="Categories.Category.BinaryProducts.html#1360" class="Function Operator">⟩</a> <a id="2086" href="Relation.Binary.Reasoning.Setoid.html#1162" class="Function">≈˘⟨</a> <a id="2090" href="Categories.Category.BinaryProducts.html#5365" class="Function">assocˡ∘⟨⟩</a> <a id="2100" href="Relation.Binary.Reasoning.Setoid.html#1162" class="Function">⟩</a>
<a id="2108" href="Categories.Category.BinaryProducts.html#1506" class="Function">assocˡ</a> <a id="2115" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="2117" href="Categories.Category.BinaryProducts.html#1360" class="Function Operator">⟨</a> <a id="2119" href="Categories.Category.BinaryProducts.html#1360" class="Function Operator">⟨</a> <a id="2121" href="Categories.Object.Product.Core.html#546" class="Function">π₂</a> <a id="2124" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="2126" href="Categories.Object.Product.Core.html#526" class="Function">π₁</a> <a id="2129" href="Categories.Category.BinaryProducts.html#1360" class="Function Operator">,</a> <a id="2131" href="Categories.Object.Product.Core.html#546" class="Function">π₂</a> <a id="2134" href="Categories.Category.BinaryProducts.html#1360" class="Function Operator">⟩</a> <a id="2136" href="Categories.Category.BinaryProducts.html#1360" class="Function Operator">,</a> <a id="2138" href="Categories.Object.Product.Core.html#526" class="Function">π₁</a> <a id="2141" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="2143" href="Categories.Object.Product.Core.html#526" class="Function">π₁</a> <a id="2146" href="Categories.Category.BinaryProducts.html#1360" class="Function Operator">⟩</a> <a id="2166" href="Relation.Binary.Reasoning.Setoid.html#1162" class="Function">≈˘⟨</a> <a id="2170" href="Categories.Category.Core.html#2734" class="Function Operator">refl⟩∘⟨</a> <a id="2178" href="Categories.Category.BinaryProducts.html#4237" class="Function">swap∘⟨⟩</a> <a id="2186" href="Relation.Binary.Reasoning.Setoid.html#1162" class="Function">⟩</a>
<a id="2194" href="Categories.Category.BinaryProducts.html#1506" class="Function">assocˡ</a> <a id="2201" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="2203" href="Categories.Category.BinaryProducts.html#2012" class="Function">swap</a> <a id="2208" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="2210" href="Categories.Category.BinaryProducts.html#1506" class="Function">assocˡ</a> <a id="2252" href="Relation.Binary.Reasoning.Base.Single.html#2564" class="Function Operator">∎</a>
<a id="symmetric"></a><a id="2255" href="Categories.Category.Cartesian.SymmetricMonoidal.html#2255" class="Function">symmetric</a> <a id="2265" class="Symbol">:</a> <a id="2267" href="Categories.Category.Monoidal.Symmetric.html#997" class="Record">Symmetric</a>
<a id="2277" href="Categories.Category.Cartesian.SymmetricMonoidal.html#2255" class="Function">symmetric</a> <a id="2287" class="Symbol">=</a> <a id="2289" href="Categories.Category.Monoidal.Symmetric.html#3126" class="Function">symmetricHelper</a> <a id="2305" class="Keyword">record</a>
<a id="2314" class="Symbol">{</a> <a id="2316" href="Categories.Category.Monoidal.Symmetric.html#1583" class="Field">braiding</a> <a id="2328" class="Symbol">=</a> <a id="2330" class="Keyword">record</a>
<a id="2341" class="Symbol">{</a> <a id="2343" href="Categories.NaturalTransformation.NaturalIsomorphism.html#891" class="Field">F⇒G</a> <a id="2347" class="Symbol">=</a> <a id="2349" href="Categories.NaturalTransformation.Core.html#1750" class="Function">ntHelper</a> <a id="2358" class="Keyword">record</a>
<a id="2371" class="Symbol">{</a> <a id="2373" href="Categories.NaturalTransformation.Core.html#1637" class="Field">η</a> <a id="2381" class="Symbol">=</a> <a id="2383" class="Symbol">λ</a> <a id="2385" href="Categories.Category.Cartesian.SymmetricMonoidal.html#2385" class="Bound">_</a> <a id="2387" class="Symbol">→</a> <a id="2389" href="Categories.Category.BinaryProducts.html#2012" class="Function">swap</a>
<a id="2400" class="Symbol">;</a> <a id="2402" href="Categories.NaturalTransformation.Core.html#1681" class="Field">commute</a> <a id="2410" class="Symbol">=</a> <a id="2412" class="Symbol">λ</a> <a id="2414" href="Categories.Category.Cartesian.SymmetricMonoidal.html#2414" class="Bound">_</a> <a id="2416" class="Symbol">→</a> <a id="2418" href="Categories.Category.BinaryProducts.html#4476" class="Function">swap∘⁂</a>
<a id="2431" class="Symbol">}</a>
<a id="2437" class="Symbol">;</a> <a id="2439" href="Categories.NaturalTransformation.NaturalIsomorphism.html#927" class="Field">F⇐G</a> <a id="2443" class="Symbol">=</a> <a id="2445" href="Categories.NaturalTransformation.Core.html#1750" class="Function">ntHelper</a> <a id="2454" class="Keyword">record</a>
<a id="2467" class="Symbol">{</a> <a id="2469" href="Categories.NaturalTransformation.Core.html#1637" class="Field">η</a> <a id="2477" class="Symbol">=</a> <a id="2479" class="Symbol">λ</a> <a id="2481" href="Categories.Category.Cartesian.SymmetricMonoidal.html#2481" class="Bound">_</a> <a id="2483" class="Symbol">→</a> <a id="2485" href="Categories.Category.BinaryProducts.html#2012" class="Function">swap</a>
<a id="2496" class="Symbol">;</a> <a id="2498" href="Categories.NaturalTransformation.Core.html#1681" class="Field">commute</a> <a id="2506" class="Symbol">=</a> <a id="2508" class="Symbol">λ</a> <a id="2510" href="Categories.Category.Cartesian.SymmetricMonoidal.html#2510" class="Bound">_</a> <a id="2512" class="Symbol">→</a> <a id="2514" href="Categories.Category.BinaryProducts.html#4476" class="Function">swap∘⁂</a>
<a id="2527" class="Symbol">}</a>
<a id="2533" class="Symbol">;</a> <a id="2535" href="Categories.NaturalTransformation.NaturalIsomorphism.html#1051" class="Field">iso</a> <a id="2539" class="Symbol">=</a> <a id="2541" class="Symbol">λ</a> <a id="2543" href="Categories.Category.Cartesian.SymmetricMonoidal.html#2543" class="Bound">_</a> <a id="2545" class="Symbol">→</a> <a id="2547" class="Keyword">record</a>
<a id="2560" class="Symbol">{</a> <a id="2562" href="Categories.Morphism.html#1586" class="Field">isoˡ</a> <a id="2567" class="Symbol">=</a> <a id="2569" href="Categories.Category.BinaryProducts.html#4652" class="Function">swap∘swap</a>
<a id="2585" class="Symbol">;</a> <a id="2587" href="Categories.Morphism.html#1612" class="Field">isoʳ</a> <a id="2592" class="Symbol">=</a> <a id="2594" href="Categories.Category.BinaryProducts.html#4652" class="Function">swap∘swap</a>
<a id="2610" class="Symbol">}</a>
<a id="2616" class="Symbol">}</a>
<a id="2620" class="Symbol">;</a> <a id="2622" href="Categories.Category.Monoidal.Symmetric.html#1790" class="Field">commutative</a> <a id="2634" class="Symbol">=</a> <a id="2636" href="Categories.Category.BinaryProducts.html#4652" class="Function">swap∘swap</a>
<a id="2648" class="Symbol">;</a> <a id="2650" href="Categories.Category.Monoidal.Symmetric.html#1837" class="Field">hexagon</a> <a id="2662" class="Symbol">=</a> <a id="2664" href="Categories.Category.Cartesian.SymmetricMonoidal.html#1219" class="Function">hexagon</a>
<a id="2674" class="Symbol">}</a>
</pre></body></html>
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