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<html><head><meta charset="utf-8"><title>Categories.Category.Cartesian.Monoidal</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="107" class="Comment">-- Defines the induced Monoidal structure of a Cartesian Category</a>
<a id="174" class="Keyword">module</a> <a id="181" href="Categories.Category.Cartesian.Monoidal.html" class="Module">Categories.Category.Cartesian.Monoidal</a> <a id="220" class="Symbol">{</a><a id="221" href="Categories.Category.Cartesian.Monoidal.html#221" class="Bound">o</a> <a id="223" href="Categories.Category.Cartesian.Monoidal.html#223" class="Bound">ℓ</a> <a id="225" href="Categories.Category.Cartesian.Monoidal.html#225" class="Bound">e</a><a id="226" class="Symbol">}</a> <a id="228" class="Symbol">{</a><a id="229" href="Categories.Category.Cartesian.Monoidal.html#229" class="Bound">𝒞</a> <a id="231" class="Symbol">:</a> <a id="233" href="Categories.Category.Core.html#442" class="Record">Category</a> <a id="242" href="Categories.Category.Cartesian.Monoidal.html#221" class="Bound">o</a> <a id="244" href="Categories.Category.Cartesian.Monoidal.html#223" class="Bound">ℓ</a> <a id="246" href="Categories.Category.Cartesian.Monoidal.html#225" class="Bound">e</a><a id="247" class="Symbol">}</a> <a id="249" class="Keyword">where</a>
<a id="256" class="Keyword">open</a> <a id="261" href="Categories.Category.Core.html#442" class="Module">Category</a> <a id="270" href="Categories.Category.Cartesian.Monoidal.html#229" class="Bound">𝒞</a>
<a id="272" class="Keyword">open</a> <a id="277" href="Categories.Category.Core.html#2462" class="Module">HomReasoning</a>
<a id="291" class="Keyword">open</a> <a id="296" class="Keyword">import</a> <a id="303" href="Categories.Category.BinaryProducts.html" class="Module">Categories.Category.BinaryProducts</a> <a id="338" href="Categories.Category.Cartesian.Monoidal.html#229" class="Bound">𝒞</a> <a id="340" class="Keyword">using</a> <a id="346" class="Symbol">(</a><a id="347" href="Categories.Category.BinaryProducts.html#812" class="Record">BinaryProducts</a><a id="361" class="Symbol">;</a> <a id="363" class="Keyword">module</a> <a id="370" href="Categories.Category.BinaryProducts.html#812" class="Module">BinaryProducts</a><a id="384" class="Symbol">)</a>
<a id="386" class="Keyword">open</a> <a id="391" class="Keyword">import</a> <a id="398" href="Categories.Category.Cartesian.html" class="Module">Categories.Category.Cartesian</a> <a id="428" class="Keyword">using</a> <a id="434" class="Symbol">(</a><a id="435" href="Categories.Category.Cartesian.html#727" class="Record">Cartesian</a><a id="444" class="Symbol">)</a>
<a id="446" class="Keyword">open</a> <a id="451" class="Keyword">import</a> <a id="458" href="Categories.Object.Terminal.html" class="Module">Categories.Object.Terminal</a> <a id="485" href="Categories.Category.Cartesian.Monoidal.html#229" class="Bound">𝒞</a> <a id="487" class="Keyword">using</a> <a id="493" class="Symbol">(</a><a id="494" href="Categories.Object.Terminal.html#860" class="Record">Terminal</a><a id="502" class="Symbol">)</a>
<a id="504" class="Keyword">open</a> <a id="509" class="Keyword">import</a> <a id="516" href="Categories.Object.Product.Core.html" class="Module">Categories.Object.Product.Core</a> <a id="547" href="Categories.Category.Cartesian.Monoidal.html#229" class="Bound">𝒞</a> <a id="549" class="Keyword">using</a> <a id="555" class="Symbol">(</a><a id="556" class="Keyword">module</a> <a id="563" href="Categories.Object.Product.Core.html#435" class="Module">Product</a><a id="570" class="Symbol">)</a>
<a id="572" class="Keyword">open</a> <a id="577" class="Keyword">import</a> <a id="584" href="Categories.Morphism.html" class="Module">Categories.Morphism</a> <a id="604" href="Categories.Category.Cartesian.Monoidal.html#229" class="Bound">𝒞</a> <a id="606" class="Keyword">using</a> <a id="612" class="Symbol">(</a><a id="613" href="Categories.Morphism.html#1958" class="Record Operator">_≅_</a><a id="616" class="Symbol">;</a> <a id="618" class="Keyword">module</a> <a id="625" href="Categories.Morphism.html#3269" class="Module">≅</a><a id="626" class="Symbol">)</a>
<a id="628" class="Keyword">open</a> <a id="633" class="Keyword">import</a> <a id="640" href="Categories.Morphism.Reasoning.html" class="Module">Categories.Morphism.Reasoning</a> <a id="670" href="Categories.Category.Cartesian.Monoidal.html#229" class="Bound">𝒞</a> <a id="672" class="Keyword">using</a> <a id="678" class="Symbol">(</a><a id="679" href="Categories.Morphism.Reasoning.Core.html#6851" class="Function">cancelˡ</a><a id="686" class="Symbol">;</a> <a id="688" href="Categories.Morphism.Reasoning.Core.html#1914" class="Function">pullʳ</a><a id="693" class="Symbol">;</a> <a id="695" href="Categories.Morphism.Reasoning.Core.html#2048" class="Function">pullˡ</a><a id="700" class="Symbol">)</a>
<a id="702" class="Keyword">open</a> <a id="707" class="Keyword">import</a> <a id="714" href="Categories.Category.Monoidal.html" class="Module">Categories.Category.Monoidal</a> <a id="743" class="Keyword">using</a> <a id="749" class="Symbol">(</a><a id="750" href="Categories.Category.Monoidal.Core.html#1287" class="Record">Monoidal</a><a id="758" class="Symbol">)</a>
<a id="761" class="Keyword">open</a> <a id="766" class="Keyword">import</a> <a id="773" href="Categories.Functor.html" class="Module">Categories.Functor</a> <a id="792" class="Keyword">using</a> <a id="798" class="Symbol">(</a><a id="799" href="Categories.Functor.Core.html#248" class="Record">Functor</a><a id="806" class="Symbol">)</a> <a id="808" class="Keyword">renaming</a> <a id="817" class="Symbol">(</a><a id="818" href="Categories.Functor.html#349" class="Function">id</a> <a id="821" class="Symbol">to</a> <a id="824" class="Function">idF</a><a id="827" class="Symbol">)</a>
<a id="829" class="Keyword">open</a> <a id="834" class="Keyword">import</a> <a id="841" href="Categories.NaturalTransformation.html" class="Module">Categories.NaturalTransformation</a> <a id="874" class="Keyword">using</a> <a id="880" class="Symbol">(</a><a id="881" href="Categories.NaturalTransformation.Core.html#1750" class="Function">ntHelper</a><a id="889" class="Symbol">)</a>
<a id="891" class="Keyword">open</a> <a id="896" class="Keyword">import</a> <a id="903" href="Categories.NaturalTransformation.NaturalIsomorphism.html" class="Module">Categories.NaturalTransformation.NaturalIsomorphism</a> <a id="955" class="Keyword">using</a> <a id="961" class="Symbol">(</a><a id="962" href="Categories.NaturalTransformation.NaturalIsomorphism.html#651" class="Record">NaturalIsomorphism</a><a id="980" class="Symbol">)</a>
<a id="983" class="Keyword">private</a>
<a id="993" class="Keyword">variable</a>
<a id="1006" href="Categories.Category.Cartesian.Monoidal.html#1006" class="Generalizable">A</a> <a id="1008" href="Categories.Category.Cartesian.Monoidal.html#1008" class="Generalizable">B</a> <a id="1010" href="Categories.Category.Cartesian.Monoidal.html#1010" class="Generalizable">C</a> <a id="1012" href="Categories.Category.Cartesian.Monoidal.html#1012" class="Generalizable">D</a> <a id="1014" href="Categories.Category.Cartesian.Monoidal.html#1014" class="Generalizable">W</a> <a id="1016" href="Categories.Category.Cartesian.Monoidal.html#1016" class="Generalizable">X</a> <a id="1018" href="Categories.Category.Cartesian.Monoidal.html#1018" class="Generalizable">Y</a> <a id="1020" href="Categories.Category.Cartesian.Monoidal.html#1020" class="Generalizable">Z</a> <a id="1022" class="Symbol">:</a> <a id="1024" href="Categories.Category.Core.html#559" class="Field">Obj</a>
<a id="1032" href="Categories.Category.Cartesian.Monoidal.html#1032" class="Generalizable">f</a> <a id="1034" href="Categories.Category.Cartesian.Monoidal.html#1034" class="Generalizable">f′</a> <a id="1037" href="Categories.Category.Cartesian.Monoidal.html#1037" class="Generalizable">g</a> <a id="1039" href="Categories.Category.Cartesian.Monoidal.html#1039" class="Generalizable">g′</a> <a id="1042" href="Categories.Category.Cartesian.Monoidal.html#1042" class="Generalizable">h</a> <a id="1044" href="Categories.Category.Cartesian.Monoidal.html#1044" class="Generalizable">i</a> <a id="1046" class="Symbol">:</a> <a id="1048" href="Categories.Category.Cartesian.Monoidal.html#1006" class="Generalizable">A</a> <a id="1050" href="Categories.Category.Core.html#575" class="Field Operator">⇒</a> <a id="1052" href="Categories.Category.Cartesian.Monoidal.html#1008" class="Generalizable">B</a>
<a id="1055" class="Comment">-- The cartesian structure induces a monoidal one: 𝒞 is cartesian monoidal.</a>
<a id="1132" class="Keyword">module</a> <a id="CartesianMonoidal"></a><a id="1139" href="Categories.Category.Cartesian.Monoidal.html#1139" class="Module">CartesianMonoidal</a> <a id="1157" class="Symbol">(</a><a id="1158" href="Categories.Category.Cartesian.Monoidal.html#1158" class="Bound">cartesian</a> <a id="1168" class="Symbol">:</a> <a id="1170" href="Categories.Category.Cartesian.html#727" class="Record">Cartesian</a> <a id="1180" href="Categories.Category.Cartesian.Monoidal.html#229" class="Bound">𝒞</a><a id="1181" class="Symbol">)</a> <a id="1183" class="Keyword">where</a>
<a id="1191" class="Keyword">open</a> <a id="1196" href="Categories.Category.html#1298" class="Module">Commutation</a> <a id="1208" href="Categories.Category.Cartesian.Monoidal.html#229" class="Bound">𝒞</a>
<a id="1212" class="Keyword">open</a> <a id="1217" href="Categories.Object.Terminal.html#860" class="Module">Terminal</a> <a id="1226" class="Symbol">(</a><a id="1227" href="Categories.Category.Cartesian.html#777" class="Field">Cartesian.terminal</a> <a id="1246" href="Categories.Category.Cartesian.Monoidal.html#1158" class="Bound">cartesian</a><a id="1255" class="Symbol">)</a> <a id="1258" class="Keyword">using</a> <a id="1264" class="Symbol">(</a><a id="1265" href="Categories.Object.Terminal.html#905" class="Field">⊤</a><a id="1266" class="Symbol">;</a> <a id="1268" href="Categories.Object.Terminal.html#577" class="Function">!</a><a id="1269" class="Symbol">;</a> <a id="1271" href="Categories.Object.Terminal.html#605" class="Function">!-unique</a><a id="1279" class="Symbol">;</a> <a id="1281" href="Categories.Object.Terminal.html#647" class="Function">!-unique₂</a><a id="1290" class="Symbol">)</a>
<a id="1294" class="Keyword">open</a> <a id="1299" href="Categories.Category.BinaryProducts.html#812" class="Module">BinaryProducts</a> <a id="1314" class="Symbol">(</a><a id="1315" href="Categories.Category.Cartesian.html#801" class="Field">Cartesian.products</a> <a id="1334" href="Categories.Category.Cartesian.Monoidal.html#1158" class="Bound">cartesian</a><a id="1343" class="Symbol">)</a> <a id="1345" class="Keyword">using</a> <a id="1351" class="Symbol">(</a><a id="1352" href="Categories.Object.Product.Core.html#526" class="Function">π₁</a><a id="1354" class="Symbol">;</a> <a id="1356" href="Categories.Object.Product.Core.html#546" class="Function">π₂</a><a id="1358" class="Symbol">;</a> <a id="1360" href="Categories.Category.BinaryProducts.html#1360" class="Function Operator">⟨_,_⟩</a><a id="1365" class="Symbol">;</a> <a id="1367" href="Categories.Category.BinaryProducts.html#1017" class="Function Operator">_×_</a><a id="1370" class="Symbol">;</a> <a id="1372" href="Categories.Category.BinaryProducts.html#1429" class="Function Operator">_⁂_</a><a id="1375" class="Symbol">;</a>
<a id="1381" href="Categories.Category.BinaryProducts.html#7350" class="Function Operator">_×-</a><a id="1384" class="Symbol">;</a> <a id="1386" href="Categories.Category.BinaryProducts.html#7306" class="Function Operator">-×_</a><a id="1389" class="Symbol">;</a> <a id="1391" href="Categories.Category.BinaryProducts.html#2939" class="Function">⟨⟩∘</a><a id="1394" class="Symbol">;</a> <a id="1396" href="Categories.Object.Product.Core.html#854" class="Function">⟨⟩-cong₂</a><a id="1404" class="Symbol">;</a> <a id="1406" href="Categories.Category.BinaryProducts.html#7074" class="Function">-×-</a><a id="1409" class="Symbol">;</a> <a id="1411" href="Categories.Category.BinaryProducts.html#1147" class="Function">×-assoc</a><a id="1418" class="Symbol">;</a> <a id="1420" href="Categories.Category.BinaryProducts.html#6161" class="Function">assocˡ∘⁂</a><a id="1428" class="Symbol">;</a> <a id="1430" href="Categories.Category.BinaryProducts.html#5644" class="Function">assocʳ∘⁂</a><a id="1438" class="Symbol">;</a> <a id="1440" href="Categories.Category.BinaryProducts.html#2563" class="Function">⁂∘⟨⟩</a><a id="1444" class="Symbol">;</a>
<a id="1450" href="Categories.Category.BinaryProducts.html#2652" class="Function">first∘⟨⟩</a><a id="1458" class="Symbol">;</a> <a id="1460" href="Categories.Category.BinaryProducts.html#2747" class="Function">second∘⟨⟩</a><a id="1469" class="Symbol">;</a> <a id="1471" href="Categories.Category.BinaryProducts.html#1927" class="Function">⟨⟩-congˡ</a><a id="1479" class="Symbol">;</a> <a id="1481" href="Categories.Category.BinaryProducts.html#1842" class="Function">⟨⟩-congʳ</a><a id="1489" class="Symbol">;</a> <a id="1491" href="Categories.Category.BinaryProducts.html#2355" class="Function">π₁∘⁂</a><a id="1495" class="Symbol">;</a> <a id="1497" href="Categories.Category.BinaryProducts.html#2417" class="Function">π₂∘⁂</a><a id="1501" class="Symbol">;</a> <a id="1503" href="Categories.Category.BinaryProducts.html#5365" class="Function">assocˡ∘⟨⟩</a><a id="1512" class="Symbol">;</a>
<a id="1518" href="Categories.Category.BinaryProducts.html#1506" class="Function">assocˡ</a><a id="1524" class="Symbol">;</a> <a id="1526" href="Categories.Category.BinaryProducts.html#1568" class="Function">assocʳ</a><a id="1532" class="Symbol">;</a>
<a id="1538" href="Categories.Object.Product.Core.html#797" class="Function">η</a><a id="1539" class="Symbol">;</a> <a id="1541" href="Categories.Object.Product.Core.html#671" class="Function">unique</a><a id="1547" class="Symbol">;</a> <a id="1549" href="Categories.Object.Product.Core.html#603" class="Function">project₁</a><a id="1557" class="Symbol">;</a> <a id="1559" href="Categories.Object.Product.Core.html#637" class="Function">project₂</a><a id="1567" class="Symbol">)</a>
<a id="CartesianMonoidal.⊤×A≅A"></a><a id="1572" href="Categories.Category.Cartesian.Monoidal.html#1572" class="Function">⊤×A≅A</a> <a id="1578" class="Symbol">:</a> <a id="1580" href="Categories.Object.Terminal.html#905" class="Function">⊤</a> <a id="1582" href="Categories.Category.BinaryProducts.html#1017" class="Function Operator">×</a> <a id="1584" href="Categories.Category.Cartesian.Monoidal.html#1006" class="Generalizable">A</a> <a id="1586" href="Categories.Morphism.html#1958" class="Record Operator">≅</a> <a id="1588" href="Categories.Category.Cartesian.Monoidal.html#1006" class="Generalizable">A</a>
<a id="1592" href="Categories.Category.Cartesian.Monoidal.html#1572" class="Function">⊤×A≅A</a> <a id="1598" class="Symbol">=</a> <a id="1600" class="Keyword">record</a>
<a id="1611" class="Symbol">{</a> <a id="1613" href="Categories.Morphism.html#2006" class="Field">from</a> <a id="1618" class="Symbol">=</a> <a id="1620" href="Categories.Object.Product.Core.html#546" class="Function">π₂</a>
<a id="1627" class="Symbol">;</a> <a id="1629" href="Categories.Morphism.html#2023" class="Field">to</a> <a id="1634" class="Symbol">=</a> <a id="1636" href="Categories.Category.BinaryProducts.html#1360" class="Function Operator">⟨</a> <a id="1638" href="Categories.Object.Terminal.html#577" class="Function">!</a> <a id="1640" href="Categories.Category.BinaryProducts.html#1360" class="Function Operator">,</a> <a id="1642" href="Categories.Category.Core.html#630" class="Field">id</a> <a id="1645" href="Categories.Category.BinaryProducts.html#1360" class="Function Operator">⟩</a>
<a id="1651" class="Symbol">;</a> <a id="1653" href="Categories.Morphism.html#2040" class="Field">iso</a> <a id="1658" class="Symbol">=</a> <a id="1660" class="Keyword">record</a>
<a id="1673" class="Symbol">{</a> <a id="1675" href="Categories.Morphism.html#1586" class="Field">isoˡ</a> <a id="1680" class="Symbol">=</a> <a id="1682" href="Relation.Binary.Reasoning.Base.Single.html#1925" class="Function Operator">begin</a>
<a id="1696" href="Categories.Category.BinaryProducts.html#1360" class="Function Operator">⟨</a> <a id="1698" href="Categories.Object.Terminal.html#577" class="Function">!</a> <a id="1700" href="Categories.Category.BinaryProducts.html#1360" class="Function Operator">,</a> <a id="1702" href="Categories.Category.Core.html#630" class="Field">id</a> <a id="1705" href="Categories.Category.BinaryProducts.html#1360" class="Function Operator">⟩</a> <a id="1707" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="1709" href="Categories.Object.Product.Core.html#546" class="Function">π₂</a> <a id="1712" href="Relation.Binary.Reasoning.Setoid.html#1162" class="Function">≈˘⟨</a> <a id="1716" href="Categories.Object.Product.Core.html#671" class="Function">unique</a> <a id="1723" href="Categories.Object.Terminal.html#647" class="Function">!-unique₂</a> <a id="1733" class="Symbol">(</a><a id="1734" href="Categories.Morphism.Reasoning.Core.html#6851" class="Function">cancelˡ</a> <a id="1742" href="Categories.Object.Product.Core.html#637" class="Function">project₂</a><a id="1750" class="Symbol">)</a> <a id="1752" href="Relation.Binary.Reasoning.Setoid.html#1162" class="Function">⟩</a>
<a id="1762" href="Categories.Category.BinaryProducts.html#1360" class="Function Operator">⟨</a> <a id="1764" href="Categories.Object.Product.Core.html#526" class="Function">π₁</a> <a id="1767" href="Categories.Category.BinaryProducts.html#1360" class="Function Operator">,</a> <a id="1769" href="Categories.Object.Product.Core.html#546" class="Function">π₂</a> <a id="1772" href="Categories.Category.BinaryProducts.html#1360" class="Function Operator">⟩</a> <a id="1778" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">≈⟨</a> <a id="1781" href="Categories.Object.Product.Core.html#797" class="Function">η</a> <a id="1783" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">⟩</a>
<a id="1793" href="Categories.Category.Core.html#630" class="Field">id</a> <a id="1809" href="Relation.Binary.Reasoning.Base.Single.html#2564" class="Function Operator">∎</a>
<a id="1817" class="Symbol">;</a> <a id="1819" href="Categories.Morphism.html#1612" class="Field">isoʳ</a> <a id="1824" class="Symbol">=</a> <a id="1826" href="Categories.Object.Product.Core.html#637" class="Function">project₂</a>
<a id="1841" class="Symbol">}</a>
<a id="1847" class="Symbol">}</a>
<a id="CartesianMonoidal.A×⊤≅A"></a><a id="1852" href="Categories.Category.Cartesian.Monoidal.html#1852" class="Function">A×⊤≅A</a> <a id="1858" class="Symbol">:</a> <a id="1860" href="Categories.Category.Cartesian.Monoidal.html#1006" class="Generalizable">A</a> <a id="1862" href="Categories.Category.BinaryProducts.html#1017" class="Function Operator">×</a> <a id="1864" href="Categories.Object.Terminal.html#905" class="Function">⊤</a> <a id="1866" href="Categories.Morphism.html#1958" class="Record Operator">≅</a> <a id="1868" href="Categories.Category.Cartesian.Monoidal.html#1006" class="Generalizable">A</a>
<a id="1872" href="Categories.Category.Cartesian.Monoidal.html#1852" class="Function">A×⊤≅A</a> <a id="1878" class="Symbol">=</a> <a id="1880" class="Keyword">record</a>
<a id="1891" class="Symbol">{</a> <a id="1893" href="Categories.Morphism.html#2006" class="Field">from</a> <a id="1898" class="Symbol">=</a> <a id="1900" href="Categories.Object.Product.Core.html#526" class="Function">π₁</a>
<a id="1907" class="Symbol">;</a> <a id="1909" href="Categories.Morphism.html#2023" class="Field">to</a> <a id="1914" class="Symbol">=</a> <a id="1916" href="Categories.Category.BinaryProducts.html#1360" class="Function Operator">⟨</a> <a id="1918" href="Categories.Category.Core.html#630" class="Field">id</a> <a id="1921" href="Categories.Category.BinaryProducts.html#1360" class="Function Operator">,</a> <a id="1923" href="Categories.Object.Terminal.html#577" class="Function">!</a> <a id="1925" href="Categories.Category.BinaryProducts.html#1360" class="Function Operator">⟩</a>
<a id="1931" class="Symbol">;</a> <a id="1933" href="Categories.Morphism.html#2040" class="Field">iso</a> <a id="1938" class="Symbol">=</a> <a id="1940" class="Keyword">record</a>
<a id="1953" class="Symbol">{</a> <a id="1955" href="Categories.Morphism.html#1586" class="Field">isoˡ</a> <a id="1960" class="Symbol">=</a> <a id="1962" href="Relation.Binary.Reasoning.Base.Single.html#1925" class="Function Operator">begin</a>
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<a id="4449" href="Categories.Category.BinaryProducts.html#1360" class="Function Operator">⟨</a> <a id="4451" href="Categories.Object.Product.Core.html#526" class="Function">π₁</a> <a id="4454" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="4456" href="Categories.Category.Cartesian.Monoidal.html#3800" class="Function">α⇒</a> <a id="4459" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="4461" href="Categories.Object.Product.Core.html#526" class="Function">π₁</a> <a id="4464" href="Categories.Category.BinaryProducts.html#1360" class="Function Operator">,</a> <a id="4466" href="Categories.Category.Cartesian.Monoidal.html#3800" class="Function">α⇒</a> <a id="4469" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="4471" href="Categories.Category.BinaryProducts.html#1360" class="Function Operator">⟨</a> <a id="4473" class="Symbol">(</a><a id="4474" href="Categories.Object.Product.Core.html#546" class="Function">π₂</a> <a id="4477" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="4479" href="Categories.Object.Product.Core.html#526" class="Function">π₁</a><a id="4481" class="Symbol">)</a> <a id="4483" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="4485" class="Symbol">(</a><a id="4486" href="Categories.Category.Cartesian.Monoidal.html#3800" class="Function">α⇒</a> <a id="4489" href="Categories.Category.BinaryProducts.html#1429" class="Function Operator">⁂</a> <a id="4491" href="Categories.Category.Core.html#630" class="Field">id</a><a id="4493" class="Symbol">)</a> <a id="4495" href="Categories.Category.BinaryProducts.html#1360" class="Function Operator">,</a> <a id="4497" href="Categories.Object.Product.Core.html#546" class="Function">π₂</a> <a id="4500" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="4502" class="Symbol">(</a><a id="4503" href="Categories.Category.Cartesian.Monoidal.html#3800" class="Function">α⇒</a> <a id="4506" href="Categories.Category.BinaryProducts.html#1429" class="Function Operator">⁂</a> <a id="4508" href="Categories.Category.Core.html#630" class="Field">id</a><a id="4510" class="Symbol">)</a> <a id="4512" href="Categories.Category.BinaryProducts.html#1360" class="Function Operator">⟩</a> <a id="4514" href="Categories.Category.BinaryProducts.html#1360" class="Function Operator">⟩</a> <a id="4517" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">≈⟨</a> <a id="4520" href="Categories.Object.Product.Core.html#854" class="Function">⟨⟩-cong₂</a> <a id="4529" class="Symbol">(</a><a id="4530" href="Categories.Morphism.Reasoning.Core.html#2048" class="Function">pullˡ</a> <a id="4536" href="Categories.Object.Product.Core.html#603" class="Function">project₁</a><a id="4544" class="Symbol">)</a> <a id="4546" class="Symbol">(</a> <a id="4548" href="Categories.Category.Core.html#2734" class="Function Operator">refl⟩∘⟨</a> <a id="4556" href="Categories.Object.Product.Core.html#854" class="Function">⟨⟩-cong₂</a> <a id="4565" class="Symbol">(</a><a id="4566" href="Categories.Morphism.Reasoning.Core.html#1914" class="Function">pullʳ</a> <a id="4572" href="Categories.Category.BinaryProducts.html#2355" class="Function">π₁∘⁂</a><a id="4576" class="Symbol">)</a> <a id="4578" href="Categories.Category.BinaryProducts.html#2417" class="Function">π₂∘⁂</a><a id="4582" class="Symbol">)</a> <a id="4584" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">⟩</a>
<a id="4592" href="Categories.Category.BinaryProducts.html#1360" class="Function Operator">⟨</a> <a id="4594" class="Symbol">(</a><a id="4595" href="Categories.Object.Product.Core.html#526" class="Function">π₁</a> <a id="4598" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="4600" href="Categories.Object.Product.Core.html#526" class="Function">π₁</a><a id="4602" class="Symbol">)</a> <a id="4604" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="4606" href="Categories.Object.Product.Core.html#526" class="Function">π₁</a> <a id="4609" href="Categories.Category.BinaryProducts.html#1360" class="Function Operator">,</a> <a id="4611" href="Categories.Category.Cartesian.Monoidal.html#3800" class="Function">α⇒</a> <a id="4614" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="4616" href="Categories.Category.BinaryProducts.html#1360" class="Function Operator">⟨</a> <a id="4618" href="Categories.Object.Product.Core.html#546" class="Function">π₂</a> <a id="4621" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="4623" href="Categories.Category.Cartesian.Monoidal.html#3800" class="Function">α⇒</a> <a id="4626" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="4628" href="Categories.Object.Product.Core.html#526" class="Function">π₁</a> <a id="4631" href="Categories.Category.BinaryProducts.html#1360" class="Function Operator">,</a> <a id="4633" href="Categories.Category.Core.html#630" class="Field">id</a> <a id="4636" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="4638" href="Categories.Object.Product.Core.html#546" class="Function">π₂</a> <a id="4641" href="Categories.Category.BinaryProducts.html#1360" class="Function Operator">⟩</a> <a id="4643" href="Categories.Category.BinaryProducts.html#1360" class="Function Operator">⟩</a> <a id="4660" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">≈⟨</a> <a id="4663" href="Categories.Object.Product.Core.html#854" class="Function">⟨⟩-cong₂</a> <a id="4672" href="Categories.Category.Core.html#715" class="Field">assoc</a> <a id="4678" class="Symbol">(</a><a id="4679" href="Categories.Category.Core.html#2734" class="Function Operator">refl⟩∘⟨</a> <a id="4687" href="Categories.Object.Product.Core.html#854" class="Function">⟨⟩-cong₂</a> <a id="4696" class="Symbol">(</a><a id="4697" href="Categories.Morphism.Reasoning.Core.html#2048" class="Function">pullˡ</a> <a id="4703" href="Categories.Object.Product.Core.html#637" class="Function">project₂</a><a id="4711" class="Symbol">)</a> <a id="4713" href="Categories.Category.Core.html#1096" class="Field">identityˡ</a><a id="4722" class="Symbol">)</a> <a id="4724" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">⟩</a>
<a id="4732" href="Categories.Category.BinaryProducts.html#1360" class="Function Operator">⟨</a> <a id="4734" href="Categories.Category.Cartesian.Monoidal.html#5377" class="Function">π₁₁₁</a> <a id="4739" href="Categories.Category.BinaryProducts.html#1360" class="Function Operator">,</a> <a id="4741" href="Categories.Category.Cartesian.Monoidal.html#3800" class="Function">α⇒</a> <a id="4744" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="4746" href="Categories.Category.BinaryProducts.html#1360" class="Function Operator">⟨</a> <a id="4748" href="Categories.Category.BinaryProducts.html#1360" class="Function Operator">⟨</a> <a id="4750" href="Categories.Object.Product.Core.html#546" class="Function">π₂</a> <a id="4753" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="4755" href="Categories.Object.Product.Core.html#526" class="Function">π₁</a> <a id="4758" href="Categories.Category.BinaryProducts.html#1360" class="Function Operator">,</a> <a id="4760" href="Categories.Object.Product.Core.html#546" class="Function">π₂</a> <a id="4763" href="Categories.Category.BinaryProducts.html#1360" class="Function Operator">⟩</a> <a id="4765" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="4767" href="Categories.Object.Product.Core.html#526" class="Function">π₁</a> <a id="4770" href="Categories.Category.BinaryProducts.html#1360" class="Function Operator">,</a> <a id="4772" href="Categories.Object.Product.Core.html#546" class="Function">π₂</a> <a id="4775" href="Categories.Category.BinaryProducts.html#1360" class="Function Operator">⟩</a> <a id="4777" href="Categories.Category.BinaryProducts.html#1360" class="Function Operator">⟩</a> <a id="4801" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">≈⟨</a> <a id="4804" href="Categories.Category.BinaryProducts.html#1927" class="Function">⟨⟩-congˡ</a> <a id="4813" class="Symbol">(</a><a id="4814" href="Categories.Category.Core.html#2734" class="Function Operator">refl⟩∘⟨</a> <a id="4822" href="Categories.Category.BinaryProducts.html#1842" class="Function">⟨⟩-congʳ</a> <a id="4831" href="Categories.Category.BinaryProducts.html#2939" class="Function">⟨⟩∘</a><a id="4834" class="Symbol">)</a> <a id="4836" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">⟩</a>
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<a id="4943" href="Categories.Category.BinaryProducts.html#1360" class="Function Operator">⟨</a> <a id="4945" href="Categories.Category.Cartesian.Monoidal.html#5377" class="Function">π₁₁₁</a> <a id="4950" href="Categories.Category.BinaryProducts.html#1360" class="Function Operator">,</a> <a id="4952" href="Categories.Category.BinaryProducts.html#1360" class="Function Operator">⟨</a> <a id="4954" class="Symbol">(</a><a id="4955" href="Categories.Object.Product.Core.html#546" class="Function">π₂</a> <a id="4958" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="4960" href="Categories.Object.Product.Core.html#526" class="Function">π₁</a><a id="4962" class="Symbol">)</a> <a id="4964" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="4966" href="Categories.Object.Product.Core.html#526" class="Function">π₁</a> <a id="4969" href="Categories.Category.BinaryProducts.html#1360" class="Function Operator">,</a> <a id="4971" href="Categories.Category.BinaryProducts.html#1360" class="Function Operator">⟨</a> <a id="4973" href="Categories.Object.Product.Core.html#546" class="Function">π₂</a> <a id="4976" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="4978" href="Categories.Object.Product.Core.html#526" class="Function">π₁</a> <a id="4981" href="Categories.Category.BinaryProducts.html#1360" class="Function Operator">,</a> <a id="4983" href="Categories.Object.Product.Core.html#546" class="Function">π₂</a> <a id="4986" href="Categories.Category.BinaryProducts.html#1360" class="Function Operator">⟩</a> <a id="4988" href="Categories.Category.BinaryProducts.html#1360" class="Function Operator">⟩</a> <a id="4990" href="Categories.Category.BinaryProducts.html#1360" class="Function Operator">⟩</a> <a id="5013" href="Relation.Binary.Reasoning.Setoid.html#1162" class="Function">≈˘⟨</a> <a id="5017" href="Categories.Category.BinaryProducts.html#1927" class="Function">⟨⟩-congˡ</a> <a id="5026" class="Symbol">(</a><a id="5027" href="Categories.Object.Product.Core.html#854" class="Function">⟨⟩-cong₂</a> <a id="5036" class="Symbol">(</a><a id="5037" href="Relation.Binary.Structures.html#1629" class="Function">Equiv.trans</a> <a id="5049" class="Symbol">(</a><a id="5050" href="Categories.Morphism.Reasoning.Core.html#1914" class="Function">pullʳ</a> <a id="5056" href="Categories.Object.Product.Core.html#603" class="Function">project₁</a><a id="5064" class="Symbol">)</a> <a id="5066" href="Categories.Category.Core.html#1004" class="Field">sym-assoc</a><a id="5075" class="Symbol">)</a> <a id="5077" href="Categories.Object.Product.Core.html#637" class="Function">project₂</a><a id="5085" class="Symbol">)</a> <a id="5087" href="Relation.Binary.Reasoning.Setoid.html#1162" class="Function">⟩</a>
<a id="5095" href="Categories.Category.BinaryProducts.html#1360" class="Function Operator">⟨</a> <a id="5097" href="Categories.Category.Cartesian.Monoidal.html#5377" class="Function">π₁₁₁</a> <a id="5102" href="Categories.Category.BinaryProducts.html#1360" class="Function Operator">,</a> <a id="5104" href="Categories.Category.BinaryProducts.html#1360" class="Function Operator">⟨</a> <a id="5106" class="Symbol">(</a><a id="5107" href="Categories.Object.Product.Core.html#546" class="Function">π₂</a> <a id="5110" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="5112" href="Categories.Object.Product.Core.html#526" class="Function">π₁</a><a id="5114" class="Symbol">)</a> <a id="5116" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="5118" href="Categories.Category.Cartesian.Monoidal.html#3800" class="Function">α⇒</a> <a id="5121" href="Categories.Category.BinaryProducts.html#1360" class="Function Operator">,</a> <a id="5123" href="Categories.Object.Product.Core.html#546" class="Function">π₂</a> <a id="5126" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="5128" href="Categories.Category.Cartesian.Monoidal.html#3800" class="Function">α⇒</a> <a id="5131" href="Categories.Category.BinaryProducts.html#1360" class="Function Operator">⟩</a> <a id="5133" href="Categories.Category.BinaryProducts.html#1360" class="Function Operator">⟩</a> <a id="5163" href="Relation.Binary.Reasoning.Setoid.html#1162" class="Function">≈˘⟨</a> <a id="5167" href="Categories.Object.Product.Core.html#854" class="Function">⟨⟩-cong₂</a> <a id="5176" class="Symbol">(</a><a id="5177" href="Categories.Morphism.Reasoning.Core.html#1914" class="Function">pullʳ</a> <a id="5183" href="Categories.Object.Product.Core.html#603" class="Function">project₁</a><a id="5191" class="Symbol">)</a> <a id="5193" href="Categories.Category.BinaryProducts.html#2939" class="Function">⟨⟩∘</a> <a id="5197" href="Relation.Binary.Reasoning.Setoid.html#1162" class="Function">⟩</a>
<a id="5205" href="Categories.Category.BinaryProducts.html#1360" class="Function Operator">⟨</a> <a id="5207" class="Symbol">(</a><a id="5208" href="Categories.Object.Product.Core.html#526" class="Function">π₁</a> <a id="5211" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="5213" href="Categories.Object.Product.Core.html#526" class="Function">π₁</a><a id="5215" class="Symbol">)</a> <a id="5217" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="5219" href="Categories.Category.Cartesian.Monoidal.html#3800" class="Function">α⇒</a> <a id="5222" href="Categories.Category.BinaryProducts.html#1360" class="Function Operator">,</a> <a id="5224" href="Categories.Category.BinaryProducts.html#1360" class="Function Operator">⟨</a> <a id="5226" href="Categories.Object.Product.Core.html#546" class="Function">π₂</a> <a id="5229" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="5231" href="Categories.Object.Product.Core.html#526" class="Function">π₁</a> <a id="5234" href="Categories.Category.BinaryProducts.html#1360" class="Function Operator">,</a> <a id="5236" href="Categories.Object.Product.Core.html#546" class="Function">π₂</a> <a id="5239" href="Categories.Category.BinaryProducts.html#1360" class="Function Operator">⟩</a> <a id="5241" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="5243" href="Categories.Category.Cartesian.Monoidal.html#3800" class="Function">α⇒</a> <a id="5246" href="Categories.Category.BinaryProducts.html#1360" class="Function Operator">⟩</a> <a id="5273" href="Relation.Binary.Reasoning.Setoid.html#1162" class="Function">≈˘⟨</a> <a id="5277" href="Categories.Category.BinaryProducts.html#2939" class="Function">⟨⟩∘</a> <a id="5281" href="Relation.Binary.Reasoning.Setoid.html#1162" class="Function">⟩</a>
<a id="5289" href="Categories.Category.Cartesian.Monoidal.html#3800" class="Function">α⇒</a> <a id="5292" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="5294" href="Categories.Category.Cartesian.Monoidal.html#3800" class="Function">α⇒</a> <a id="5355" href="Relation.Binary.Reasoning.Base.Single.html#2564" class="Function Operator">∎</a>
<a id="5363" class="Keyword">where</a>
<a id="5377" href="Categories.Category.Cartesian.Monoidal.html#5377" class="Function">π₁₁₁</a> <a id="5382" class="Symbol">=</a> <a id="5384" href="Categories.Object.Product.Core.html#526" class="Function">π₁</a> <a id="5387" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="5389" href="Categories.Object.Product.Core.html#526" class="Function">π₁</a> <a id="5392" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="5394" href="Categories.Object.Product.Core.html#526" class="Function">π₁</a>
<a id="CartesianMonoidal.monoidal"></a><a id="5400" href="Categories.Category.Cartesian.Monoidal.html#5400" class="Function">monoidal</a> <a id="5409" class="Symbol">:</a> <a id="5411" href="Categories.Category.Monoidal.Core.html#1287" class="Record">Monoidal</a> <a id="5420" href="Categories.Category.Cartesian.Monoidal.html#229" class="Bound">𝒞</a>
<a id="5424" href="Categories.Category.Cartesian.Monoidal.html#5400" class="Function">monoidal</a> <a id="5433" class="Symbol">=</a> <a id="5435" class="Keyword">record</a>
<a id="5446" class="Symbol">{</a> <a id="5448" href="Categories.Category.Monoidal.Core.html#1355" class="Field">⊗</a> <a id="5469" class="Symbol">=</a> <a id="5471" href="Categories.Category.BinaryProducts.html#7074" class="Function">-×-</a>
<a id="5479" class="Symbol">;</a> <a id="5481" href="Categories.Category.Monoidal.Core.html#1380" class="Field">unit</a> <a id="5502" class="Symbol">=</a> <a id="5504" href="Categories.Object.Terminal.html#905" class="Function">⊤</a>
<a id="5510" class="Symbol">;</a> <a id="5512" href="Categories.Category.Monoidal.Core.html#1700" class="Field">unitorˡ</a> <a id="5533" class="Symbol">=</a> <a id="5535" href="Categories.Category.Cartesian.Monoidal.html#1572" class="Function">⊤×A≅A</a>
<a id="5545" class="Symbol">;</a> <a id="5547" href="Categories.Category.Monoidal.Core.html#1731" class="Field">unitorʳ</a> <a id="5568" class="Symbol">=</a> <a id="5570" href="Categories.Category.Cartesian.Monoidal.html#1852" class="Function">A×⊤≅A</a>
<a id="5580" class="Symbol">;</a> <a id="5582" href="Categories.Category.Monoidal.Core.html#1762" class="Field">associator</a> <a id="5603" class="Symbol">=</a> <a id="5605" href="Relation.Binary.Structures.html#1603" class="Function">≅.sym</a> <a id="5611" href="Categories.Category.BinaryProducts.html#1147" class="Function">×-assoc</a>
<a id="5623" class="Symbol">;</a> <a id="5625" href="Categories.Category.Monoidal.Core.html#2517" class="Field">unitorˡ-commute-from</a> <a id="5646" class="Symbol">=</a> <a id="5648" href="Categories.Object.Product.Core.html#637" class="Function">project₂</a>
<a id="5661" class="Symbol">;</a> <a id="5663" href="Categories.Category.Monoidal.Core.html#2582" class="Field">unitorˡ-commute-to</a> <a id="5684" class="Symbol">=</a> <a id="5686" class="Keyword">let</a> <a id="5690" class="Keyword">open</a> <a id="5695" href="Categories.NaturalTransformation.NaturalIsomorphism.html#651" class="Module">NaturalIsomorphism</a> <a id="5714" href="Categories.Category.Cartesian.Monoidal.html#2132" class="Function">⊤×--id</a> <a id="5721" class="Keyword">in</a> <a id="5724" href="Categories.NaturalTransformation.Core.html#827" class="Function">⇐.commute</a> <a id="5734" class="Symbol">_</a>
<a id="5740" class="Symbol">;</a> <a id="5742" href="Categories.Category.Monoidal.Core.html#2647" class="Field">unitorʳ-commute-from</a> <a id="5763" class="Symbol">=</a> <a id="5765" href="Categories.Object.Product.Core.html#603" class="Function">project₁</a>
<a id="5778" class="Symbol">;</a> <a id="5780" href="Categories.Category.Monoidal.Core.html#2712" class="Field">unitorʳ-commute-to</a> <a id="5801" class="Symbol">=</a> <a id="5803" class="Keyword">let</a> <a id="5807" class="Keyword">open</a> <a id="5812" href="Categories.NaturalTransformation.NaturalIsomorphism.html#651" class="Module">NaturalIsomorphism</a> <a id="5831" href="Categories.Category.Cartesian.Monoidal.html#2927" class="Function">-×⊤-id</a> <a id="5838" class="Keyword">in</a> <a id="5841" href="Categories.NaturalTransformation.Core.html#827" class="Function">⇐.commute</a> <a id="5851" class="Symbol">_</a>
<a id="5857" class="Symbol">;</a> <a id="5859" href="Categories.Category.Monoidal.Core.html#2777" class="Field">assoc-commute-from</a> <a id="5880" class="Symbol">=</a> <a id="5882" href="Categories.Category.BinaryProducts.html#6161" class="Function">assocˡ∘⁂</a>
<a id="5895" class="Symbol">;</a> <a id="5897" href="Categories.Category.Monoidal.Core.html#2860" class="Field">assoc-commute-to</a> <a id="5918" class="Symbol">=</a> <a id="5920" href="Categories.Category.BinaryProducts.html#5644" class="Function">assocʳ∘⁂</a>
<a id="5933" class="Symbol">;</a> <a id="5935" href="Categories.Category.Monoidal.Core.html#2943" class="Field">triangle</a> <a id="5956" class="Symbol">=</a> <a id="5958" href="Relation.Binary.Reasoning.Base.Single.html#1925" class="Function Operator">begin</a>
<a id="5970" class="Symbol">(</a><a id="5971" href="Categories.Category.Core.html#630" class="Field">id</a> <a id="5974" href="Categories.Category.BinaryProducts.html#1429" class="Function Operator">⁂</a> <a id="5976" href="Categories.Object.Product.Core.html#546" class="Function">π₂</a><a id="5978" class="Symbol">)</a> <a id="5980" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="5982" href="Categories.Category.BinaryProducts.html#1506" class="Function">assocˡ</a> <a id="6011" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">≈⟨</a> <a id="6014" href="Categories.Category.BinaryProducts.html#2563" class="Function">⁂∘⟨⟩</a> <a id="6019" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">⟩</a>
<a id="6027" href="Categories.Category.BinaryProducts.html#1360" class="Function Operator">⟨</a> <a id="6029" href="Categories.Category.Core.html#630" class="Field">id</a> <a id="6032" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="6034" href="Categories.Object.Product.Core.html#526" class="Function">π₁</a> <a id="6037" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="6039" href="Categories.Object.Product.Core.html#526" class="Function">π₁</a> <a id="6042" href="Categories.Category.BinaryProducts.html#1360" class="Function Operator">,</a> <a id="6044" href="Categories.Object.Product.Core.html#546" class="Function">π₂</a> <a id="6047" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="6049" href="Categories.Category.BinaryProducts.html#1360" class="Function Operator">⟨</a> <a id="6051" href="Categories.Object.Product.Core.html#546" class="Function">π₂</a> <a id="6054" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="6056" href="Categories.Object.Product.Core.html#526" class="Function">π₁</a> <a id="6059" href="Categories.Category.BinaryProducts.html#1360" class="Function Operator">,</a> <a id="6061" href="Categories.Object.Product.Core.html#546" class="Function">π₂</a> <a id="6064" href="Categories.Category.BinaryProducts.html#1360" class="Function Operator">⟩</a> <a id="6066" href="Categories.Category.BinaryProducts.html#1360" class="Function Operator">⟩</a> <a id="6068" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">≈⟨</a> <a id="6071" href="Categories.Object.Product.Core.html#854" class="Function">⟨⟩-cong₂</a> <a id="6080" class="Symbol">(</a><a id="6081" href="Categories.Morphism.Reasoning.Core.html#2048" class="Function">pullˡ</a> <a id="6087" href="Categories.Category.Core.html#1096" class="Field">identityˡ</a><a id="6096" class="Symbol">)</a> <a id="6098" class="Symbol">(</a><a id="6099" href="Categories.Object.Product.Core.html#637" class="Function">project₂</a> <a id="6108" href="Categories.Category.Core.html#3061" class="Function Operator">○</a> <a id="6110" class="Symbol">(</a><a id="6111" href="Categories.Category.Core.html#3005" class="Function">⟺</a> <a id="6113" href="Categories.Category.Core.html#1096" class="Field">identityˡ</a><a id="6122" class="Symbol">))</a> <a id="6125" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">⟩</a>
<a id="6133" href="Categories.Object.Product.Core.html#526" class="Function">π₁</a> <a id="6136" href="Categories.Category.BinaryProducts.html#1429" class="Function Operator">⁂</a> <a id="6138" href="Categories.Category.Core.html#630" class="Field">id</a> <a id="6174" href="Relation.Binary.Reasoning.Base.Single.html#2564" class="Function Operator">∎</a>
<a id="6180" class="Symbol">;</a> <a id="6182" href="Categories.Category.Monoidal.Core.html#3174" class="Field">pentagon</a> <a id="6203" class="Symbol">=</a> <a id="6205" href="Categories.Category.Cartesian.Monoidal.html#3828" class="Function">pentagon</a>
<a id="6218" class="Symbol">}</a>
</pre></body></html>
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