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<html><head><meta charset="utf-8"><title>Categories.Functor.Duality</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">module</a> <a id="44" href="Categories.Functor.Duality.html" class="Module">Categories.Functor.Duality</a> <a id="71" class="Keyword">where</a>
<a id="78" class="Keyword">open</a> <a id="83" class="Keyword">import</a> <a id="90" href="Level.html" class="Module">Level</a>
<a id="96" class="Keyword">open</a> <a id="101" class="Keyword">import</a> <a id="108" href="Data.Product.html" class="Module">Data.Product</a> <a id="121" class="Keyword">using</a> <a id="127" class="Symbol">(</a><a id="128" href="Agda.Builtin.Sigma.html#165" class="Record">Σ</a><a id="129" class="Symbol">;</a> <a id="131" href="Agda.Builtin.Sigma.html#235" class="InductiveConstructor Operator">_,_</a><a id="134" class="Symbol">)</a>
<a id="137" class="Keyword">open</a> <a id="142" class="Keyword">import</a> <a id="149" href="Categories.Category.html" class="Module">Categories.Category</a>
<a id="169" class="Keyword">open</a> <a id="174" class="Keyword">import</a> <a id="181" href="Categories.Category.Construction.Cones.html" class="Module">Categories.Category.Construction.Cones</a> <a id="220" class="Symbol">as</a> <a id="223" class="Module">Con</a>
<a id="227" class="Keyword">open</a> <a id="232" class="Keyword">import</a> <a id="239" href="Categories.Category.Construction.Cocones.html" class="Module">Categories.Category.Construction.Cocones</a> <a id="280" class="Symbol">as</a> <a id="283" class="Module">Coc</a>
<a id="287" class="Keyword">open</a> <a id="292" class="Keyword">import</a> <a id="299" href="Categories.Functor.html" class="Module">Categories.Functor</a>
<a id="318" class="Keyword">open</a> <a id="323" class="Keyword">import</a> <a id="330" href="Categories.Functor.Limits.html" class="Module">Categories.Functor.Limits</a>
<a id="356" class="Keyword">open</a> <a id="361" class="Keyword">import</a> <a id="368" href="Categories.Object.Initial.html" class="Module">Categories.Object.Initial</a>
<a id="394" class="Keyword">open</a> <a id="399" class="Keyword">import</a> <a id="406" href="Categories.Object.Terminal.html" class="Module">Categories.Object.Terminal</a>
<a id="433" class="Keyword">open</a> <a id="438" class="Keyword">import</a> <a id="445" href="Categories.Diagram.Limit.html" class="Module">Categories.Diagram.Limit</a> <a id="470" class="Symbol">as</a> <a id="473" class="Module">Lim</a>
<a id="477" class="Keyword">open</a> <a id="482" class="Keyword">import</a> <a id="489" href="Categories.Diagram.Colimit.html" class="Module">Categories.Diagram.Colimit</a> <a id="516" class="Symbol">as</a> <a id="519" class="Module">Col</a>
<a id="523" class="Keyword">open</a> <a id="528" class="Keyword">import</a> <a id="535" href="Categories.Diagram.Duality.html" class="Module">Categories.Diagram.Duality</a>
<a id="562" class="Keyword">open</a> <a id="567" class="Keyword">import</a> <a id="574" href="Categories.Morphism.Duality.html" class="Module">Categories.Morphism.Duality</a> <a id="602" class="Symbol">as</a> <a id="605" class="Module">MorD</a>
<a id="611" class="Keyword">open</a> <a id="616" class="Keyword">import</a> <a id="623" href="Categories.Functor.Algebra.html" class="Module">Categories.Functor.Algebra</a> <a id="650" class="Keyword">using</a> <a id="656" class="Symbol">(</a><a id="657" href="Categories.Functor.Algebra.html#487" class="Record">F-Algebra</a><a id="666" class="Symbol">;</a> <a id="668" href="Categories.Functor.Algebra.html#1307" class="Record">F-Algebra-Morphism</a><a id="686" class="Symbol">)</a>
<a id="688" class="Keyword">open</a> <a id="693" class="Keyword">import</a> <a id="700" href="Categories.Functor.Coalgebra.html" class="Module">Categories.Functor.Coalgebra</a> <a id="729" class="Keyword">using</a> <a id="735" class="Symbol">(</a><a id="736" href="Categories.Functor.Coalgebra.html#483" class="Record">F-Coalgebra</a><a id="747" class="Symbol">;</a> <a id="749" href="Categories.Functor.Coalgebra.html#1350" class="Record">F-Coalgebra-Morphism</a><a id="769" class="Symbol">)</a>
<a id="771" class="Keyword">open</a> <a id="776" class="Keyword">import</a> <a id="783" href="Relation.Binary.PropositionalEquality.html" class="Module">Relation.Binary.PropositionalEquality</a> <a id="821" class="Keyword">using</a> <a id="827" class="Symbol">(</a><a id="828" href="Agda.Builtin.Equality.html#150" class="Datatype Operator">_≡_</a><a id="831" class="Symbol">;</a> <a id="833" href="Agda.Builtin.Equality.html#207" class="InductiveConstructor">refl</a><a id="837" class="Symbol">)</a>
<a id="840" class="Keyword">private</a>
<a id="850" class="Keyword">variable</a>
<a id="863" href="Categories.Functor.Duality.html#863" class="Generalizable">o</a> <a id="865" href="Categories.Functor.Duality.html#865" class="Generalizable">ℓ</a> <a id="867" href="Categories.Functor.Duality.html#867" class="Generalizable">e</a> <a id="869" class="Symbol">:</a> <a id="871" href="Agda.Primitive.html#742" class="Postulate">Level</a>
<a id="881" href="Categories.Functor.Duality.html#881" class="Generalizable">C</a> <a id="883" href="Categories.Functor.Duality.html#883" class="Generalizable">D</a> <a id="885" href="Categories.Functor.Duality.html#885" class="Generalizable">E</a> <a id="887" href="Categories.Functor.Duality.html#887" class="Generalizable">J</a> <a id="889" class="Symbol">:</a> <a id="891" href="Categories.Category.Core.html#442" class="Record">Category</a> <a id="900" href="Categories.Functor.Duality.html#863" class="Generalizable">o</a> <a id="902" href="Categories.Functor.Duality.html#865" class="Generalizable">ℓ</a> <a id="904" href="Categories.Functor.Duality.html#867" class="Generalizable">e</a>
<a id="907" class="Keyword">module</a> <a id="914" href="Categories.Functor.Duality.html#914" class="Module">_</a> <a id="916" class="Symbol">(</a><a id="917" href="Categories.Functor.Duality.html#917" class="Bound">G</a> <a id="919" class="Symbol">:</a> <a id="921" href="Categories.Functor.Core.html#248" class="Record">Functor</a> <a id="929" href="Categories.Functor.Duality.html#881" class="Generalizable">C</a> <a id="931" href="Categories.Functor.Duality.html#883" class="Generalizable">D</a><a id="932" class="Symbol">)</a> <a id="934" class="Symbol">{</a><a id="935" href="Categories.Functor.Duality.html#935" class="Bound">J</a> <a id="937" class="Symbol">:</a> <a id="939" href="Categories.Category.Core.html#442" class="Record">Category</a> <a id="948" href="Categories.Functor.Duality.html#863" class="Generalizable">o</a> <a id="950" href="Categories.Functor.Duality.html#865" class="Generalizable">ℓ</a> <a id="952" href="Categories.Functor.Duality.html#867" class="Generalizable">e</a><a id="953" class="Symbol">}</a> <a id="955" class="Keyword">where</a>
<a id="963" class="Keyword">private</a>
<a id="975" class="Keyword">module</a> <a id="982" href="Categories.Functor.Duality.html#982" class="Module">C</a> <a id="984" class="Symbol">=</a> <a id="986" href="Categories.Category.Core.html#442" class="Module">Category</a> <a id="995" href="Categories.Functor.Duality.html#929" class="Bound">C</a>
<a id="1001" class="Keyword">module</a> <a id="1008" href="Categories.Functor.Duality.html#1008" class="Module">D</a> <a id="1010" class="Symbol">=</a> <a id="1012" href="Categories.Category.Core.html#442" class="Module">Category</a> <a id="1021" href="Categories.Functor.Duality.html#931" class="Bound">D</a>
<a id="1027" class="Keyword">module</a> <a id="1034" href="Categories.Functor.Duality.html#1034" class="Module">G</a> <a id="1036" class="Symbol">=</a> <a id="1038" href="Categories.Functor.Core.html#248" class="Module">Functor</a> <a id="1046" href="Categories.Functor.Duality.html#917" class="Bound">G</a>
<a id="1052" class="Keyword">module</a> <a id="1059" href="Categories.Functor.Duality.html#1059" class="Module">J</a> <a id="1061" class="Symbol">=</a> <a id="1063" href="Categories.Category.Core.html#442" class="Module">Category</a> <a id="1072" href="Categories.Functor.Duality.html#935" class="Bound">J</a>
<a id="1077" href="Categories.Functor.Duality.html#1077" class="Function">coPreservesLimit⇒PreservesCoLimit</a> <a id="1111" class="Symbol">:</a> <a id="1113" class="Symbol">∀</a> <a id="1115" class="Symbol">{</a><a id="1116" href="Categories.Functor.Duality.html#1116" class="Bound">F</a> <a id="1118" class="Symbol">:</a> <a id="1120" href="Categories.Functor.Core.html#248" class="Record">Functor</a> <a id="1128" href="Categories.Functor.Duality.html#935" class="Bound">J</a> <a id="1130" href="Categories.Functor.Duality.html#929" class="Bound">C</a><a id="1131" class="Symbol">}</a> <a id="1133" class="Symbol">(</a><a id="1134" href="Categories.Functor.Duality.html#1134" class="Bound">L</a> <a id="1136" class="Symbol">:</a> <a id="1138" href="Categories.Diagram.Limit.html#934" class="Record">Limit</a> <a id="1144" class="Symbol">(</a><a id="1145" href="Categories.Functor.Core.html#816" class="Function">Functor.op</a> <a id="1156" href="Categories.Functor.Duality.html#1116" class="Bound">F</a><a id="1157" class="Symbol">))</a> <a id="1160" class="Symbol">→</a>
<a id="1204" href="Categories.Functor.Limits.html#675" class="Function">PreservesLimit</a> <a id="1219" href="Categories.Functor.Core.html#816" class="Function">G.op</a> <a id="1224" href="Categories.Functor.Duality.html#1134" class="Bound">L</a> <a id="1226" class="Symbol">→</a>
<a id="1270" href="Categories.Functor.Limits.html#811" class="Function">PreservesColimit</a> <a id="1287" href="Categories.Functor.Duality.html#917" class="Bound">G</a> <a id="1289" class="Symbol">(</a><a id="1290" href="Categories.Diagram.Duality.html#4031" class="Function">coLimit⇒Colimit</a> <a id="1306" href="Categories.Functor.Duality.html#929" class="Bound">C</a> <a id="1308" href="Categories.Functor.Duality.html#1134" class="Bound">L</a><a id="1309" class="Symbol">)</a>
<a id="1313" href="Categories.Functor.Duality.html#1077" class="Function">coPreservesLimit⇒PreservesCoLimit</a> <a id="1347" href="Categories.Functor.Duality.html#1347" class="Bound">L</a> <a id="1349" href="Categories.Functor.Duality.html#1349" class="Bound">is⊤</a> <a id="1353" class="Symbol">=</a> <a id="1355" class="Keyword">record</a>
<a id="1366" class="Symbol">{</a> <a id="1368" href="Categories.Object.Initial.html#478" class="Field">!</a> <a id="1377" class="Symbol">=</a> <a id="1379" class="Symbol">λ</a> <a id="1381" class="Symbol">{</a><a id="1382" href="Categories.Functor.Duality.html#1382" class="Bound">K</a><a id="1383" class="Symbol">}</a> <a id="1385" class="Symbol">→</a> <a id="1387" href="Categories.Diagram.Duality.html#3633" class="Function">coCone⇒⇒Cocone⇒</a> <a id="1403" class="Symbol">_</a> <a id="1405" class="Symbol">(</a><a id="1406" href="Categories.Object.Terminal.html#577" class="Field">!</a> <a id="1408" class="Symbol">{</a><a id="1409" href="Categories.Diagram.Duality.html#3494" class="Function">Cocone⇒coCone</a> <a id="1423" class="Symbol">_</a> <a id="1425" href="Categories.Functor.Duality.html#1382" class="Bound">K</a><a id="1426" class="Symbol">})</a>
<a id="1433" class="Symbol">;</a> <a id="1435" href="Categories.Object.Initial.html#506" class="Field">!-unique</a> <a id="1444" class="Symbol">=</a> <a id="1446" class="Symbol">λ</a> <a id="1448" href="Categories.Functor.Duality.html#1448" class="Bound">f</a> <a id="1450" class="Symbol">→</a> <a id="1452" href="Categories.Object.Terminal.html#605" class="Field">!-unique</a> <a id="1461" class="Symbol">(</a><a id="1462" href="Categories.Diagram.Duality.html#3831" class="Function">Cocone⇒⇒coCone⇒</a> <a id="1478" class="Symbol">_</a> <a id="1480" href="Categories.Functor.Duality.html#1448" class="Bound">f</a><a id="1481" class="Symbol">)</a>
<a id="1487" class="Symbol">}</a>
<a id="1493" class="Keyword">where</a> <a id="1499" class="Keyword">open</a> <a id="1504" href="Categories.Object.Terminal.html#520" class="Module">IsTerminal</a> <a id="1515" href="Categories.Functor.Duality.html#1349" class="Bound">is⊤</a>
<a id="1522" href="Categories.Functor.Duality.html#1522" class="Function">PreservesColimit⇒coPreservesLimit</a> <a id="1556" class="Symbol">:</a> <a id="1558" class="Symbol">∀</a> <a id="1560" class="Symbol">{</a><a id="1561" href="Categories.Functor.Duality.html#1561" class="Bound">F</a> <a id="1563" class="Symbol">:</a> <a id="1565" href="Categories.Functor.Core.html#248" class="Record">Functor</a> <a id="1573" href="Categories.Functor.Duality.html#935" class="Bound">J</a> <a id="1575" href="Categories.Functor.Duality.html#929" class="Bound">C</a><a id="1576" class="Symbol">}</a> <a id="1578" class="Symbol">(</a><a id="1579" href="Categories.Functor.Duality.html#1579" class="Bound">L</a> <a id="1581" class="Symbol">:</a> <a id="1583" href="Categories.Diagram.Colimit.html#961" class="Record">Colimit</a> <a id="1591" href="Categories.Functor.Duality.html#1561" class="Bound">F</a><a id="1592" class="Symbol">)</a> <a id="1594" class="Symbol">→</a>
<a id="1636" href="Categories.Functor.Limits.html#811" class="Function">PreservesColimit</a> <a id="1653" href="Categories.Functor.Duality.html#917" class="Bound">G</a> <a id="1655" href="Categories.Functor.Duality.html#1579" class="Bound">L</a> <a id="1657" class="Symbol">→</a>
<a id="1699" href="Categories.Functor.Limits.html#675" class="Function">PreservesLimit</a> <a id="1714" href="Categories.Functor.Core.html#816" class="Function">G.op</a> <a id="1719" class="Symbol">(</a><a id="1720" href="Categories.Diagram.Duality.html#4402" class="Function">Colimit⇒coLimit</a> <a id="1736" href="Categories.Functor.Duality.html#929" class="Bound">C</a> <a id="1738" href="Categories.Functor.Duality.html#1579" class="Bound">L</a><a id="1739" class="Symbol">)</a>
<a id="1743" href="Categories.Functor.Duality.html#1522" class="Function">PreservesColimit⇒coPreservesLimit</a> <a id="1777" href="Categories.Functor.Duality.html#1777" class="Bound">L</a> <a id="1779" href="Categories.Functor.Duality.html#1779" class="Bound">is⊥</a> <a id="1783" class="Symbol">=</a> <a id="1785" class="Keyword">record</a>
<a id="1796" class="Symbol">{</a> <a id="1798" href="Categories.Object.Terminal.html#577" class="Field">!</a> <a id="1807" class="Symbol">=</a> <a id="1809" class="Symbol">λ</a> <a id="1811" class="Symbol">{</a><a id="1812" href="Categories.Functor.Duality.html#1812" class="Bound">K</a><a id="1813" class="Symbol">}</a> <a id="1815" class="Symbol">→</a> <a id="1817" href="Categories.Diagram.Duality.html#3831" class="Function">Cocone⇒⇒coCone⇒</a> <a id="1833" class="Symbol">_</a> <a id="1835" class="Symbol">(</a><a id="1836" href="Categories.Object.Initial.html#478" class="Field">!</a> <a id="1838" class="Symbol">{</a><a id="1839" href="Categories.Diagram.Duality.html#3186" class="Function">coCone⇒Cocone</a> <a id="1853" class="Symbol">_</a> <a id="1855" href="Categories.Functor.Duality.html#1812" class="Bound">K</a><a id="1856" class="Symbol">})</a>
<a id="1863" class="Symbol">;</a> <a id="1865" href="Categories.Object.Terminal.html#605" class="Field">!-unique</a> <a id="1874" class="Symbol">=</a> <a id="1876" class="Symbol">λ</a> <a id="1878" href="Categories.Functor.Duality.html#1878" class="Bound">f</a> <a id="1880" class="Symbol">→</a> <a id="1882" href="Categories.Object.Initial.html#506" class="Field">!-unique</a> <a id="1891" class="Symbol">(</a><a id="1892" href="Categories.Diagram.Duality.html#3633" class="Function">coCone⇒⇒Cocone⇒</a> <a id="1908" class="Symbol">_</a> <a id="1910" href="Categories.Functor.Duality.html#1878" class="Bound">f</a><a id="1911" class="Symbol">)</a>
<a id="1917" class="Symbol">}</a>
<a id="1923" class="Keyword">where</a> <a id="1929" class="Keyword">open</a> <a id="1934" href="Categories.Object.Initial.html#422" class="Module">IsInitial</a> <a id="1944" href="Categories.Functor.Duality.html#1779" class="Bound">is⊥</a>
<a id="1949" class="Keyword">module</a> <a id="1956" href="Categories.Functor.Duality.html#1956" class="Module">_</a> <a id="1958" class="Symbol">{</a><a id="1959" href="Categories.Functor.Duality.html#1959" class="Bound">o</a> <a id="1961" href="Categories.Functor.Duality.html#1961" class="Bound">ℓ</a> <a id="1963" href="Categories.Functor.Duality.html#1963" class="Bound">e</a><a id="1964" class="Symbol">}</a> <a id="1966" class="Symbol">(</a><a id="1967" href="Categories.Functor.Duality.html#1967" class="Bound">G</a> <a id="1969" class="Symbol">:</a> <a id="1971" href="Categories.Functor.Core.html#248" class="Record">Functor</a> <a id="1979" href="Categories.Functor.Duality.html#881" class="Generalizable">C</a> <a id="1981" href="Categories.Functor.Duality.html#883" class="Generalizable">D</a><a id="1982" class="Symbol">)</a> <a id="1984" class="Keyword">where</a>
<a id="1992" class="Keyword">private</a>
<a id="2004" class="Keyword">module</a> <a id="2011" href="Categories.Functor.Duality.html#2011" class="Module">G</a> <a id="2013" class="Symbol">=</a> <a id="2015" href="Categories.Functor.Core.html#248" class="Module">Functor</a> <a id="2023" href="Categories.Functor.Duality.html#1967" class="Bound">G</a>
<a id="2028" href="Categories.Functor.Duality.html#2028" class="Function">coContinuous⇒Cocontinuous</a> <a id="2054" class="Symbol">:</a> <a id="2056" href="Categories.Functor.Limits.html#1594" class="Function">Continuous</a> <a id="2067" href="Categories.Functor.Duality.html#1959" class="Bound">o</a> <a id="2069" href="Categories.Functor.Duality.html#1961" class="Bound">ℓ</a> <a id="2071" href="Categories.Functor.Duality.html#1963" class="Bound">e</a> <a id="2073" href="Categories.Functor.Core.html#816" class="Function">G.op</a> <a id="2078" class="Symbol">→</a> <a id="2080" href="Categories.Functor.Limits.html#1749" class="Function">Cocontinuous</a> <a id="2093" href="Categories.Functor.Duality.html#1959" class="Bound">o</a> <a id="2095" href="Categories.Functor.Duality.html#1961" class="Bound">ℓ</a> <a id="2097" href="Categories.Functor.Duality.html#1963" class="Bound">e</a> <a id="2099" href="Categories.Functor.Duality.html#1967" class="Bound">G</a>
<a id="2103" href="Categories.Functor.Duality.html#2028" class="Function">coContinuous⇒Cocontinuous</a> <a id="2129" href="Categories.Functor.Duality.html#2129" class="Bound">Con</a> <a id="2133" href="Categories.Functor.Duality.html#2133" class="Bound">L</a> <a id="2135" class="Symbol">=</a>
<a id="2141" href="Categories.Functor.Duality.html#1077" class="Function">coPreservesLimit⇒PreservesCoLimit</a> <a id="2175" href="Categories.Functor.Duality.html#1967" class="Bound">G</a> <a id="2177" class="Symbol">(</a><a id="2178" href="Categories.Diagram.Duality.html#4402" class="Function">Colimit⇒coLimit</a> <a id="2194" href="Categories.Functor.Duality.html#1979" class="Bound">C</a> <a id="2196" href="Categories.Functor.Duality.html#2133" class="Bound">L</a><a id="2197" class="Symbol">)</a> <a id="2199" class="Symbol">(</a><a id="2200" href="Categories.Functor.Duality.html#2129" class="Bound">Con</a> <a id="2204" class="Symbol">(</a><a id="2205" href="Categories.Diagram.Duality.html#4402" class="Function">Colimit⇒coLimit</a> <a id="2221" href="Categories.Functor.Duality.html#1979" class="Bound">C</a> <a id="2223" href="Categories.Functor.Duality.html#2133" class="Bound">L</a><a id="2224" class="Symbol">))</a>
<a id="2230" href="Categories.Functor.Duality.html#2230" class="Function">Cocontinuous⇒coContinuous</a> <a id="2256" class="Symbol">:</a> <a id="2258" href="Categories.Functor.Limits.html#1749" class="Function">Cocontinuous</a> <a id="2271" href="Categories.Functor.Duality.html#1959" class="Bound">o</a> <a id="2273" href="Categories.Functor.Duality.html#1961" class="Bound">ℓ</a> <a id="2275" href="Categories.Functor.Duality.html#1963" class="Bound">e</a> <a id="2277" href="Categories.Functor.Duality.html#1967" class="Bound">G</a> <a id="2279" class="Symbol">→</a> <a id="2281" href="Categories.Functor.Limits.html#1594" class="Function">Continuous</a> <a id="2292" href="Categories.Functor.Duality.html#1959" class="Bound">o</a> <a id="2294" href="Categories.Functor.Duality.html#1961" class="Bound">ℓ</a> <a id="2296" href="Categories.Functor.Duality.html#1963" class="Bound">e</a> <a id="2298" href="Categories.Functor.Core.html#816" class="Function">G.op</a>
<a id="2305" href="Categories.Functor.Duality.html#2230" class="Function">Cocontinuous⇒coContinuous</a> <a id="2331" href="Categories.Functor.Duality.html#2331" class="Bound">Coc</a> <a id="2335" href="Categories.Functor.Duality.html#2335" class="Bound">L</a> <a id="2337" class="Symbol">=</a>
<a id="2343" href="Categories.Functor.Duality.html#1522" class="Function">PreservesColimit⇒coPreservesLimit</a> <a id="2377" href="Categories.Functor.Duality.html#1967" class="Bound">G</a> <a id="2379" class="Symbol">(</a><a id="2380" href="Categories.Diagram.Duality.html#4031" class="Function">coLimit⇒Colimit</a> <a id="2396" href="Categories.Functor.Duality.html#1979" class="Bound">C</a> <a id="2398" href="Categories.Functor.Duality.html#2335" class="Bound">L</a><a id="2399" class="Symbol">)</a> <a id="2401" class="Symbol">(</a><a id="2402" href="Categories.Functor.Duality.html#2331" class="Bound">Coc</a> <a id="2406" class="Symbol">(</a><a id="2407" href="Categories.Diagram.Duality.html#4031" class="Function">coLimit⇒Colimit</a> <a id="2423" href="Categories.Functor.Duality.html#1979" class="Bound">C</a> <a id="2425" href="Categories.Functor.Duality.html#2335" class="Bound">L</a><a id="2426" class="Symbol">))</a>
<a id="2430" class="Keyword">module</a> <a id="2437" href="Categories.Functor.Duality.html#2437" class="Module">_</a> <a id="2439" class="Symbol">{</a><a id="2440" href="Categories.Functor.Duality.html#2440" class="Bound">F</a> <a id="2442" class="Symbol">:</a> <a id="2444" href="Categories.Functor.html#283" class="Function">Endofunctor</a> <a id="2456" href="Categories.Functor.Duality.html#881" class="Generalizable">C</a><a id="2457" class="Symbol">}</a> <a id="2459" class="Keyword">where</a>
<a id="2467" class="Keyword">open</a> <a id="2472" href="Categories.Functor.Core.html#248" class="Module">Functor</a> <a id="2480" href="Categories.Functor.Duality.html#2440" class="Bound">F</a> <a id="2482" class="Keyword">renaming</a> <a id="2491" class="Symbol">(</a><a id="2492" href="Categories.Functor.Core.html#816" class="Function">op</a> <a id="2495" class="Symbol">to</a> <a id="2498" class="Function">Fop</a><a id="2501" class="Symbol">)</a>
<a id="2506" class="Comment">-- Obj conversions</a>
<a id="2528" href="Categories.Functor.Duality.html#2528" class="Function">coF-Algebra⇒F-Coalgebra</a> <a id="2552" class="Symbol">:</a> <a id="2554" href="Categories.Functor.Algebra.html#487" class="Record">F-Algebra</a> <a id="2564" href="Categories.Functor.Duality.html#2498" class="Function">Fop</a> <a id="2568" class="Symbol">→</a> <a id="2570" href="Categories.Functor.Coalgebra.html#483" class="Record">F-Coalgebra</a> <a id="2582" href="Categories.Functor.Duality.html#2440" class="Bound">F</a>
<a id="2586" href="Categories.Functor.Duality.html#2528" class="Function">coF-Algebra⇒F-Coalgebra</a> <a id="2610" href="Categories.Functor.Duality.html#2610" class="Bound">algOp</a> <a id="2616" class="Symbol">=</a> <a id="2618" class="Keyword">record</a> <a id="2625" class="Symbol">{</a> <a id="2627" href="Categories.Functor.Coalgebra.html#571" class="Field">A</a> <a id="2629" class="Symbol">=</a> <a id="2631" href="Categories.Functor.Algebra.html#573" class="Field">A</a> <a id="2633" href="Categories.Functor.Duality.html#2610" class="Bound">algOp</a><a id="2638" class="Symbol">;</a> <a id="2640" href="Categories.Functor.Coalgebra.html#585" class="Field">α</a> <a id="2642" class="Symbol">=</a> <a id="2644" href="Categories.Functor.Algebra.html#587" class="Field">α</a> <a id="2646" href="Categories.Functor.Duality.html#2610" class="Bound">algOp</a> <a id="2652" class="Symbol">}</a> <a id="2654" class="Keyword">where</a>
<a id="2664" class="Keyword">open</a> <a id="2669" href="Categories.Functor.Algebra.html#487" class="Module">F-Algebra</a>
<a id="2682" href="Categories.Functor.Duality.html#2682" class="Function">F-Coalgebra⇒coF-Algebra</a> <a id="2706" class="Symbol">:</a> <a id="2708" href="Categories.Functor.Coalgebra.html#483" class="Record">F-Coalgebra</a> <a id="2720" href="Categories.Functor.Duality.html#2440" class="Bound">F</a> <a id="2722" class="Symbol">→</a> <a id="2724" href="Categories.Functor.Algebra.html#487" class="Record">F-Algebra</a> <a id="2734" href="Categories.Functor.Duality.html#2498" class="Function">Fop</a>
<a id="2740" href="Categories.Functor.Duality.html#2682" class="Function">F-Coalgebra⇒coF-Algebra</a> <a id="2764" href="Categories.Functor.Duality.html#2764" class="Bound">algOp</a> <a id="2770" class="Symbol">=</a> <a id="2772" class="Keyword">record</a> <a id="2779" class="Symbol">{</a> <a id="2781" href="Categories.Functor.Algebra.html#573" class="Field">A</a> <a id="2783" class="Symbol">=</a> <a id="2785" href="Categories.Functor.Coalgebra.html#571" class="Field">A</a> <a id="2787" href="Categories.Functor.Duality.html#2764" class="Bound">algOp</a><a id="2792" class="Symbol">;</a> <a id="2794" href="Categories.Functor.Algebra.html#587" class="Field">α</a> <a id="2796" class="Symbol">=</a> <a id="2798" href="Categories.Functor.Coalgebra.html#585" class="Field">α</a> <a id="2800" href="Categories.Functor.Duality.html#2764" class="Bound">algOp</a> <a id="2806" class="Symbol">}</a> <a id="2808" class="Keyword">where</a>
<a id="2818" class="Keyword">open</a> <a id="2823" href="Categories.Functor.Coalgebra.html#483" class="Module">F-Coalgebra</a>
<a id="2838" class="Comment">-- Mor conversions</a>
<a id="2860" href="Categories.Functor.Duality.html#2860" class="Function">coF-Algebra-Morphism⇒F-Coalgebra-Morphism</a> <a id="2902" class="Symbol">:</a> <a id="2904" class="Symbol">∀</a> <a id="2906" class="Symbol">{</a><a id="2907" href="Categories.Functor.Duality.html#2907" class="Bound">X</a> <a id="2909" href="Categories.Functor.Duality.html#2909" class="Bound">Y</a> <a id="2911" class="Symbol">:</a> <a id="2913" href="Categories.Functor.Algebra.html#487" class="Record">F-Algebra</a> <a id="2923" href="Categories.Functor.Duality.html#2498" class="Function">Fop</a><a id="2926" class="Symbol">}</a> <a id="2928" class="Symbol">→</a>
<a id="2934" href="Categories.Functor.Algebra.html#1307" class="Record">F-Algebra-Morphism</a> <a id="2953" href="Categories.Functor.Duality.html#2907" class="Bound">X</a> <a id="2955" href="Categories.Functor.Duality.html#2909" class="Bound">Y</a> <a id="2957" class="Symbol">→</a>
<a id="2965" href="Categories.Functor.Coalgebra.html#1350" class="Record">F-Coalgebra-Morphism</a> <a id="2986" class="Symbol">(</a><a id="2987" href="Categories.Functor.Duality.html#2528" class="Function">coF-Algebra⇒F-Coalgebra</a> <a id="3011" href="Categories.Functor.Duality.html#2909" class="Bound">Y</a><a id="3012" class="Symbol">)</a> <a id="3014" class="Symbol">(</a><a id="3015" href="Categories.Functor.Duality.html#2528" class="Function">coF-Algebra⇒F-Coalgebra</a> <a id="3039" href="Categories.Functor.Duality.html#2907" class="Bound">X</a><a id="3040" class="Symbol">)</a>
<a id="3044" href="Categories.Functor.Duality.html#2860" class="Function">coF-Algebra-Morphism⇒F-Coalgebra-Morphism</a> <a id="3086" href="Categories.Functor.Duality.html#3086" class="Bound">morOp</a> <a id="3092" class="Symbol">=</a>
<a id="3100" class="Keyword">record</a> <a id="3107" class="Symbol">{</a> <a id="3109" href="Categories.Functor.Coalgebra.html#1411" class="Field">f</a> <a id="3111" class="Symbol">=</a> <a id="3113" href="Categories.Functor.Algebra.html#1366" class="Field">f</a> <a id="3115" href="Categories.Functor.Duality.html#3086" class="Bound">morOp</a> <a id="3121" class="Symbol">;</a> <a id="3123" href="Categories.Functor.Coalgebra.html#1433" class="Field">commutes</a> <a id="3132" class="Symbol">=</a> <a id="3134" href="Categories.Functor.Algebra.html#1388" class="Field">commutes</a> <a id="3143" href="Categories.Functor.Duality.html#3086" class="Bound">morOp</a> <a id="3149" class="Symbol">}</a> <a id="3151" class="Keyword">where</a>
<a id="3165" class="Keyword">open</a> <a id="3170" href="Categories.Functor.Algebra.html#1307" class="Module">F-Algebra-Morphism</a>
<a id="3192" href="Categories.Functor.Duality.html#3192" class="Function">F-Coalgebra-Morphism⇒coF-Algebra-Morphism</a> <a id="3234" class="Symbol">:</a> <a id="3236" class="Symbol">∀</a> <a id="3238" class="Symbol">{</a><a id="3239" href="Categories.Functor.Duality.html#3239" class="Bound">X</a> <a id="3241" href="Categories.Functor.Duality.html#3241" class="Bound">Y</a> <a id="3243" class="Symbol">:</a> <a id="3245" href="Categories.Functor.Coalgebra.html#483" class="Record">F-Coalgebra</a> <a id="3257" href="Categories.Functor.Duality.html#2440" class="Bound">F</a><a id="3258" class="Symbol">}</a> <a id="3260" class="Symbol">→</a>
<a id="3266" href="Categories.Functor.Coalgebra.html#1350" class="Record">F-Coalgebra-Morphism</a> <a id="3287" href="Categories.Functor.Duality.html#3239" class="Bound">X</a> <a id="3289" href="Categories.Functor.Duality.html#3241" class="Bound">Y</a> <a id="3291" class="Symbol">→</a>
<a id="3299" href="Categories.Functor.Algebra.html#1307" class="Record">F-Algebra-Morphism</a> <a id="3318" class="Symbol">(</a><a id="3319" href="Categories.Functor.Duality.html#2682" class="Function">F-Coalgebra⇒coF-Algebra</a> <a id="3343" href="Categories.Functor.Duality.html#3241" class="Bound">Y</a><a id="3344" class="Symbol">)</a> <a id="3346" class="Symbol">(</a><a id="3347" href="Categories.Functor.Duality.html#2682" class="Function">F-Coalgebra⇒coF-Algebra</a> <a id="3371" href="Categories.Functor.Duality.html#3239" class="Bound">X</a><a id="3372" class="Symbol">)</a>
<a id="3376" href="Categories.Functor.Duality.html#3192" class="Function">F-Coalgebra-Morphism⇒coF-Algebra-Morphism</a> <a id="3418" href="Categories.Functor.Duality.html#3418" class="Bound">morOp</a> <a id="3424" class="Symbol">=</a>
<a id="3432" class="Keyword">record</a> <a id="3439" class="Symbol">{</a> <a id="3441" href="Categories.Functor.Algebra.html#1366" class="Field">f</a> <a id="3443" class="Symbol">=</a> <a id="3445" href="Categories.Functor.Coalgebra.html#1411" class="Field">f</a> <a id="3447" href="Categories.Functor.Duality.html#3418" class="Bound">morOp</a> <a id="3453" class="Symbol">;</a> <a id="3455" href="Categories.Functor.Algebra.html#1388" class="Field">commutes</a> <a id="3464" class="Symbol">=</a> <a id="3466" href="Categories.Functor.Coalgebra.html#1433" class="Field">commutes</a> <a id="3475" href="Categories.Functor.Duality.html#3418" class="Bound">morOp</a> <a id="3481" class="Symbol">}</a> <a id="3483" class="Keyword">where</a>
<a id="3497" class="Keyword">open</a> <a id="3502" href="Categories.Functor.Coalgebra.html#1350" class="Module">F-Coalgebra-Morphism</a>
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