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<html><head><meta charset="utf-8"><title>Categories.Yoneda.Properties</title><link rel="stylesheet" href="Agda.css"></head><body><pre class="Agda"><a id="1" class="Symbol">{-#</a> <a id="5" class="Keyword">OPTIONS</a> <a id="13" class="Pragma">--without-K</a> <a id="25" class="Pragma">--safe</a> <a id="32" class="Symbol">#-}</a>
<a id="37" class="Keyword">open</a> <a id="42" class="Keyword">import</a> <a id="49" href="Level.html" class="Module">Level</a>
<a id="55" class="Keyword">open</a> <a id="60" class="Keyword">import</a> <a id="67" href="Categories.Category.html" class="Module">Categories.Category</a> <a id="87" class="Keyword">using</a> <a id="93" class="Symbol">(</a><a id="94" href="Categories.Category.Core.html#442" class="Record">Category</a><a id="102" class="Symbol">;</a> <a id="104" href="Categories.Category.html#502" class="Function Operator">_[_,_]</a><a id="110" class="Symbol">)</a>
<a id="113" class="Comment">-- Various conclusions that can be drawn from Yoneda</a>
<a id="166" class="Comment">-- over a particular Category C</a>
<a id="198" class="Keyword">module</a> <a id="205" href="Categories.Yoneda.Properties.html" class="Module">Categories.Yoneda.Properties</a> <a id="234" class="Symbol">{</a><a id="235" href="Categories.Yoneda.Properties.html#235" class="Bound">o</a> <a id="237" href="Categories.Yoneda.Properties.html#237" class="Bound">ℓ</a> <a id="239" href="Categories.Yoneda.Properties.html#239" class="Bound">e</a> <a id="241" class="Symbol">:</a> <a id="243" href="Agda.Primitive.html#591" class="Postulate">Level</a><a id="248" class="Symbol">}</a> <a id="250" class="Symbol">(</a><a id="251" href="Categories.Yoneda.Properties.html#251" class="Bound">C</a> <a id="253" class="Symbol">:</a> <a id="255" href="Categories.Category.Core.html#442" class="Record">Category</a> <a id="264" href="Categories.Yoneda.Properties.html#235" class="Bound">o</a> <a id="266" href="Categories.Yoneda.Properties.html#237" class="Bound">ℓ</a> <a id="268" href="Categories.Yoneda.Properties.html#239" class="Bound">e</a><a id="269" class="Symbol">)</a> <a id="271" class="Keyword">where</a>
<a id="278" class="Keyword">open</a> <a id="283" class="Keyword">import</a> <a id="290" href="Function.Base.html" class="Module">Function.Base</a> <a id="304" class="Keyword">using</a> <a id="310" class="Symbol">(</a><a id="311" href="Function.Base.html#1928" class="Function Operator">_$_</a><a id="314" class="Symbol">)</a>
<a id="316" class="Keyword">open</a> <a id="321" class="Keyword">import</a> <a id="328" href="Function.Bundles.html" class="Module">Function.Bundles</a> <a id="345" class="Keyword">using</a> <a id="351" class="Symbol">(</a><a id="352" href="Function.Bundles.html#5801" class="Record">Inverse</a><a id="359" class="Symbol">)</a>
<a id="361" class="Keyword">open</a> <a id="366" class="Keyword">import</a> <a id="373" href="Function.Equality.html" class="Module">Function.Equality</a> <a id="391" class="Keyword">using</a> <a id="397" class="Symbol">(</a><a id="398" href="Function.Equality.html#898" class="Record">Π</a><a id="399" class="Symbol">;</a> <a id="401" href="Function.Equality.html#1073" class="Field Operator">_⟨$⟩_</a><a id="406" class="Symbol">;</a> <a id="408" href="Function.Equality.html#1140" class="Field">cong</a><a id="412" class="Symbol">)</a>
<a id="414" class="Keyword">open</a> <a id="419" class="Keyword">import</a> <a id="426" href="Relation.Binary.Bundles.html" class="Module">Relation.Binary.Bundles</a> <a id="450" class="Keyword">using</a> <a id="456" class="Symbol">(</a><a id="457" class="Keyword">module</a> <a id="464" href="Relation.Binary.Bundles.html#1018" class="Module">Setoid</a><a id="470" class="Symbol">)</a>
<a id="472" class="Keyword">import</a> <a id="479" href="Relation.Binary.Reasoning.Setoid.html" class="Module">Relation.Binary.Reasoning.Setoid</a> <a id="512" class="Symbol">as</a> <a id="515" class="Module">SetoidR</a>
<a id="523" class="Keyword">open</a> <a id="528" class="Keyword">import</a> <a id="535" href="Data.Product.html" class="Module">Data.Product</a> <a id="548" class="Keyword">using</a> <a id="554" class="Symbol">(</a><a id="555" href="Agda.Builtin.Sigma.html#218" class="InductiveConstructor Operator">_,_</a><a id="558" class="Symbol">;</a> <a id="560" href="Agda.Builtin.Sigma.html#148" class="Record">Σ</a><a id="561" class="Symbol">)</a>
<a id="564" class="Keyword">open</a> <a id="569" class="Keyword">import</a> <a id="576" href="Categories.Category.Product.html" class="Module">Categories.Category.Product</a>
<a id="604" class="Keyword">open</a> <a id="609" class="Keyword">import</a> <a id="616" href="Categories.Category.Construction.Presheaves.html" class="Module">Categories.Category.Construction.Presheaves</a>
<a id="660" class="Keyword">open</a> <a id="665" class="Keyword">import</a> <a id="672" href="Categories.Category.Construction.Functors.html" class="Module">Categories.Category.Construction.Functors</a>
<a id="714" class="Keyword">open</a> <a id="719" class="Keyword">import</a> <a id="726" href="Categories.Category.Instance.Setoids.html" class="Module">Categories.Category.Instance.Setoids</a>
<a id="763" class="Keyword">open</a> <a id="768" class="Keyword">import</a> <a id="775" href="Categories.Functor.html" class="Module">Categories.Functor</a> <a id="794" class="Keyword">renaming</a> <a id="803" class="Symbol">(</a><a id="804" href="Categories.Functor.html#349" class="Function">id</a> <a id="807" class="Symbol">to</a> <a id="810" class="Function">idF</a><a id="813" class="Symbol">)</a>
<a id="815" class="Keyword">open</a> <a id="820" class="Keyword">import</a> <a id="827" href="Categories.Functor.Properties.html" class="Module">Categories.Functor.Properties</a> <a id="857" class="Keyword">using</a> <a id="863" class="Symbol">(</a><a id="864" href="Categories.Functor.Properties.html#1121" class="Function">Full</a><a id="868" class="Symbol">;</a> <a id="870" href="Categories.Functor.Properties.html#973" class="Function">Faithful</a><a id="878" class="Symbol">;</a> <a id="880" href="Categories.Functor.Properties.html#1428" class="Function">FullyFaithful</a><a id="893" class="Symbol">)</a>
<a id="895" class="Keyword">open</a> <a id="900" class="Keyword">import</a> <a id="907" href="Categories.Functor.Hom.html" class="Module">Categories.Functor.Hom</a> <a id="930" class="Keyword">using</a> <a id="936" class="Symbol">(</a><a id="937" class="Keyword">module</a> <a id="944" href="Categories.Functor.Hom.html#625" class="Module">Hom</a><a id="947" class="Symbol">;</a> <a id="949" href="Categories.Functor.Hom.html#3133" class="Function Operator">Hom[_][-,_]</a><a id="960" class="Symbol">;</a> <a id="962" href="Categories.Functor.Hom.html#2973" class="Function Operator">Hom[_][-,-]</a><a id="973" class="Symbol">)</a>
<a id="975" class="Keyword">open</a> <a id="980" class="Keyword">import</a> <a id="987" href="Categories.Functor.Bifunctor.html" class="Module">Categories.Functor.Bifunctor</a>
<a id="1016" class="Keyword">open</a> <a id="1021" class="Keyword">import</a> <a id="1028" href="Categories.Functor.Presheaf.html" class="Module">Categories.Functor.Presheaf</a>
<a id="1056" class="Keyword">open</a> <a id="1061" class="Keyword">import</a> <a id="1068" href="Categories.Functor.Construction.LiftSetoids.html" class="Module">Categories.Functor.Construction.LiftSetoids</a>
<a id="1112" class="Keyword">open</a> <a id="1117" class="Keyword">import</a> <a id="1124" href="Categories.NaturalTransformation.html" class="Module">Categories.NaturalTransformation</a> <a id="1157" class="Keyword">using</a> <a id="1163" class="Symbol">(</a><a id="1164" href="Categories.NaturalTransformation.Core.html#466" class="Record">NaturalTransformation</a><a id="1185" class="Symbol">;</a> <a id="1187" href="Categories.NaturalTransformation.Core.html#1750" class="Function">ntHelper</a><a id="1195" class="Symbol">)</a> <a id="1197" class="Keyword">renaming</a> <a id="1206" class="Symbol">(</a><a id="1207" href="Categories.NaturalTransformation.Core.html#2132" class="Function">id</a> <a id="1210" class="Symbol">to</a> <a id="1213" class="Function">idN</a><a id="1216" class="Symbol">)</a>
<a id="1218" class="Keyword">open</a> <a id="1223" class="Keyword">import</a> <a id="1230" href="Categories.NaturalTransformation.NaturalIsomorphism.html" class="Module">Categories.NaturalTransformation.NaturalIsomorphism</a> <a id="1282" class="Keyword">using</a> <a id="1288" class="Symbol">(</a><a id="1289" href="Categories.NaturalTransformation.NaturalIsomorphism.html#651" class="Record">NaturalIsomorphism</a><a id="1307" class="Symbol">)</a>
<a id="1309" class="Keyword">open</a> <a id="1314" class="Keyword">import</a> <a id="1321" href="Categories.Yoneda.html" class="Module">Categories.Yoneda</a>
<a id="1340" class="Keyword">import</a> <a id="1347" href="Categories.Morphism.html" class="Module">Categories.Morphism</a> <a id="1367" class="Symbol">as</a> <a id="1370" class="Module">Mor</a>
<a id="1374" class="Keyword">import</a> <a id="1381" href="Categories.Morphism.Reasoning.html" class="Module">Categories.Morphism.Reasoning</a> <a id="1411" class="Symbol">as</a> <a id="1414" class="Module">MR</a>
<a id="1417" class="Keyword">import</a> <a id="1424" href="Categories.NaturalTransformation.Hom.html" class="Module">Categories.NaturalTransformation.Hom</a> <a id="1461" class="Symbol">as</a> <a id="1464" class="Module">NT-Hom</a>
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<a id="1574" class="Keyword">open</a> <a id="1579" href="Categories.Category.Core.html#2462" class="Module">HomReasoning</a>
<a id="1592" class="Keyword">open</a> <a id="1597" href="Categories.NaturalTransformation.Core.html#466" class="Module">NaturalTransformation</a> <a id="1619" class="Keyword">using</a> <a id="1625" class="Symbol">(</a><a id="1626" href="Categories.NaturalTransformation.Core.html#783" class="Field">η</a><a id="1627" class="Symbol">;</a> <a id="1629" href="Categories.NaturalTransformation.Core.html#827" class="Field">commute</a><a id="1636" class="Symbol">)</a>
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<a id="1697" class="Keyword">private</a>
<a id="1707" class="Keyword">module</a> <a id="CE"></a><a id="1714" href="Categories.Yoneda.Properties.html#1714" class="Module">CE</a> <a id="1717" class="Symbol">=</a> <a id="1719" href="Categories.Category.Core.html#1530" class="Module">Category.Equiv</a> <a id="1734" href="Categories.Yoneda.Properties.html#251" class="Bound">C</a> <a id="1736" class="Keyword">using</a> <a id="1742" class="Symbol">(</a><a id="1743" href="Relation.Binary.Structures.html#1577" class="Function">refl</a><a id="1747" class="Symbol">)</a>
<a id="1751" class="Keyword">module</a> <a id="C"></a><a id="1758" href="Categories.Yoneda.Properties.html#1758" class="Module">C</a> <a id="1760" class="Symbol">=</a> <a id="1762" href="Categories.Category.Core.html#442" class="Module">Category</a> <a id="1771" href="Categories.Yoneda.Properties.html#251" class="Bound">C</a> <a id="1773" class="Keyword">using</a> <a id="1779" class="Symbol">(</a><a id="1780" href="Categories.Category.Core.html#3132" class="Function">op</a><a id="1782" class="Symbol">;</a> <a id="1784" href="Categories.Category.Core.html#630" class="Field">id</a><a id="1786" class="Symbol">)</a>
<a id="YoFull"></a><a id="1789" href="Categories.Yoneda.Properties.html#1789" class="Function">YoFull</a> <a id="1796" class="Symbol">:</a> <a id="1798" href="Categories.Functor.Properties.html#1121" class="Function">Full</a> <a id="1803" href="Categories.Yoneda.html#2417" class="Function">embed</a>
<a id="1809" href="Categories.Yoneda.Properties.html#1789" class="Function">YoFull</a> <a id="1816" class="Symbol">{</a><a id="1817" href="Categories.Yoneda.Properties.html#1817" class="Bound">X</a><a id="1818" class="Symbol">}</a> <a id="1820" class="Symbol">{</a><a id="1821" href="Categories.Yoneda.Properties.html#1821" class="Bound">Y</a><a id="1822" class="Symbol">}</a> <a id="1824" class="Symbol">=</a> <a id="1826" class="Keyword">record</a>
<a id="1835" class="Symbol">{</a> <a id="1837" href="Function.Surjection.html#1185" class="Field">from</a> <a id="1854" class="Symbol">=</a> <a id="1856" class="Keyword">record</a> <a id="1863" class="Symbol">{</a> <a id="1865" href="Function.Equality.html#1073" class="Field Operator">_⟨$⟩_</a> <a id="1871" class="Symbol">=</a> <a id="1873" class="Symbol">λ</a> <a id="1875" href="Categories.Yoneda.Properties.html#1875" class="Bound">ε</a> <a id="1877" class="Symbol">→</a> <a id="1879" href="Categories.NaturalTransformation.Core.html#783" class="Field">η</a> <a id="1881" href="Categories.Yoneda.Properties.html#1875" class="Bound">ε</a> <a id="1883" href="Categories.Yoneda.Properties.html#1817" class="Bound">X</a> <a id="1885" href="Function.Equality.html#1073" class="Field Operator">⟨$⟩</a> <a id="1889" href="Categories.Category.Core.html#630" class="Field">id</a> <a id="1892" class="Symbol">;</a> <a id="1894" href="Function.Equality.html#1140" class="Field">cong</a> <a id="1899" class="Symbol">=</a> <a id="1901" class="Symbol">λ</a> <a id="1903" href="Categories.Yoneda.Properties.html#1903" class="Bound">i≈j</a> <a id="1907" class="Symbol">→</a> <a id="1909" href="Categories.Yoneda.Properties.html#1903" class="Bound">i≈j</a> <a id="1913" href="Relation.Binary.Structures.html#1577" class="Function">CE.refl</a> <a id="1921" class="Symbol">}</a>
<a id="1925" class="Symbol">;</a> <a id="1927" href="Function.Surjection.html#1218" class="Field">right-inverse-of</a> <a id="1944" class="Symbol">=</a> <a id="1946" class="Symbol">λ</a> <a id="1948" href="Categories.Yoneda.Properties.html#1948" class="Bound">ε</a> <a id="1950" class="Symbol">{</a><a id="1951" href="Categories.Yoneda.Properties.html#1951" class="Bound">x</a><a id="1952" class="Symbol">}</a> <a id="1954" class="Symbol">{</a><a id="1955" href="Categories.Yoneda.Properties.html#1955" class="Bound">z</a><a id="1956" class="Symbol">}</a> <a id="1958" class="Symbol">{</a><a id="1959" href="Categories.Yoneda.Properties.html#1959" class="Bound">y</a><a id="1960" class="Symbol">}</a> <a id="1962" href="Categories.Yoneda.Properties.html#1962" class="Bound">z≈y</a> <a id="1966" class="Symbol">→</a>
<a id="1972" href="Relation.Binary.Reasoning.Base.Single.html#1925" class="Function Operator">begin</a>
<a id="1984" class="Symbol">(</a><a id="1985" href="Categories.NaturalTransformation.Core.html#783" class="Field">η</a> <a id="1987" href="Categories.Yoneda.Properties.html#1948" class="Bound">ε</a> <a id="1989" href="Categories.Yoneda.Properties.html#1817" class="Bound">X</a> <a id="1991" href="Function.Equality.html#1073" class="Field Operator">⟨$⟩</a> <a id="1995" href="Categories.Category.Core.html#630" class="Field">id</a><a id="1997" class="Symbol">)</a> <a id="1999" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="2001" href="Categories.Yoneda.Properties.html#1955" class="Bound">z</a> <a id="2008" href="Relation.Binary.Reasoning.Setoid.html#1162" class="Function">≈˘⟨</a> <a id="2012" href="Categories.Category.Core.html#1096" class="Field">identityˡ</a> <a id="2022" href="Relation.Binary.Reasoning.Setoid.html#1162" class="Function">⟩</a>
<a id="2030" href="Categories.Category.Core.html#630" class="Field">id</a> <a id="2033" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="2035" class="Symbol">(</a><a id="2036" href="Categories.NaturalTransformation.Core.html#783" class="Field">η</a> <a id="2038" href="Categories.Yoneda.Properties.html#1948" class="Bound">ε</a> <a id="2040" href="Categories.Yoneda.Properties.html#1817" class="Bound">X</a> <a id="2042" href="Function.Equality.html#1073" class="Field Operator">⟨$⟩</a> <a id="2046" href="Categories.Category.Core.html#630" class="Field">id</a><a id="2048" class="Symbol">)</a> <a id="2050" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="2052" href="Categories.Yoneda.Properties.html#1955" class="Bound">z</a> <a id="2054" href="Relation.Binary.Reasoning.Setoid.html#1162" class="Function">≈˘⟨</a> <a id="2058" href="Categories.NaturalTransformation.Core.html#827" class="Field">commute</a> <a id="2066" href="Categories.Yoneda.Properties.html#1948" class="Bound">ε</a> <a id="2068" href="Categories.Yoneda.Properties.html#1955" class="Bound">z</a> <a id="2070" href="Relation.Binary.Structures.html#1577" class="Function">CE.refl</a> <a id="2078" href="Relation.Binary.Reasoning.Setoid.html#1162" class="Function">⟩</a>
<a id="2086" href="Categories.NaturalTransformation.Core.html#783" class="Field">η</a> <a id="2088" href="Categories.Yoneda.Properties.html#1948" class="Bound">ε</a> <a id="2090" href="Categories.Yoneda.Properties.html#1951" class="Bound">x</a> <a id="2092" href="Function.Equality.html#1073" class="Field Operator">⟨$⟩</a> <a id="2096" href="Categories.Category.Core.html#630" class="Field">id</a> <a id="2099" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="2101" href="Categories.Category.Core.html#630" class="Field">id</a> <a id="2104" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="2106" href="Categories.Yoneda.Properties.html#1955" class="Bound">z</a> <a id="2110" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">≈⟨</a> <a id="2113" href="Function.Equality.html#1140" class="Field">cong</a> <a id="2118" class="Symbol">(</a><a id="2119" href="Categories.NaturalTransformation.Core.html#783" class="Field">η</a> <a id="2121" href="Categories.Yoneda.Properties.html#1948" class="Bound">ε</a> <a id="2123" href="Categories.Yoneda.Properties.html#1951" class="Bound">x</a><a id="2124" class="Symbol">)</a> <a id="2126" class="Symbol">(</a><a id="2127" href="Categories.Category.Core.html#1096" class="Field">identityˡ</a> <a id="2137" href="Categories.Category.Core.html#3061" class="Function Operator">○</a> <a id="2139" href="Categories.Category.Core.html#1096" class="Field">identityˡ</a> <a id="2149" href="Categories.Category.Core.html#3061" class="Function Operator">○</a> <a id="2151" href="Categories.Yoneda.Properties.html#1962" class="Bound">z≈y</a><a id="2154" class="Symbol">)</a> <a id="2156" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">⟩</a>
<a id="2164" href="Categories.NaturalTransformation.Core.html#783" class="Field">η</a> <a id="2166" href="Categories.Yoneda.Properties.html#1948" class="Bound">ε</a> <a id="2168" href="Categories.Yoneda.Properties.html#1951" class="Bound">x</a> <a id="2170" href="Function.Equality.html#1073" class="Field Operator">⟨$⟩</a> <a id="2174" href="Categories.Yoneda.Properties.html#1959" class="Bound">y</a> <a id="2188" href="Relation.Binary.Reasoning.Base.Single.html#2564" class="Function Operator">∎</a>
<a id="2192" class="Symbol">}</a>
<a id="YoFaithful"></a><a id="2195" href="Categories.Yoneda.Properties.html#2195" class="Function">YoFaithful</a> <a id="2206" class="Symbol">:</a> <a id="2208" href="Categories.Functor.Properties.html#973" class="Function">Faithful</a> <a id="2217" href="Categories.Yoneda.html#2417" class="Function">embed</a>
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<a id="4550" href="Categories.Category.Core.html#630" class="Field">id</a> <a id="4553" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="4555" class="Symbol">(</a><a id="4556" href="Categories.NaturalTransformation.Core.html#783" class="Field">η</a> <a id="4558" href="Categories.Yoneda.Properties.html#4368" class="Bound">α</a> <a id="4560" class="Symbol">(</a><a id="4561" href="Categories.Functor.Core.html#797" class="Function">F.₀</a> <a id="4565" href="Categories.Yoneda.Properties.html#4462" class="Bound">Y</a> <a id="4567" href="Agda.Builtin.Sigma.html#218" class="InductiveConstructor Operator">,</a> <a id="4569" href="Categories.Yoneda.Properties.html#4462" class="Bound">Y</a><a id="4570" class="Symbol">)</a> <a id="4572" href="Function.Equality.html#1073" class="Field Operator">⟨$⟩</a> <a id="4576" href="Categories.Category.Core.html#630" class="Field">id</a><a id="4578" class="Symbol">)</a> <a id="4580" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="4582" href="Categories.Functor.Core.html#806" class="Function">F.₁</a> <a id="4586" href="Categories.Yoneda.Properties.html#4465" class="Bound">f</a> <a id="4589" href="Relation.Binary.Reasoning.Setoid.html#1162" class="Function">≈˘⟨</a> <a id="4593" href="Level.html#479" class="Field">lower</a> <a id="4599" class="Symbol">(</a><a id="4600" href="Categories.NaturalTransformation.Core.html#827" class="Function">yoneda.⇒.commute</a> <a id="4617" class="Symbol">{</a><a id="4618" class="Argument">Y</a> <a id="4620" class="Symbol">=</a> <a id="4622" href="Categories.Functor.Hom.html#3133" class="Function Operator">Hom[</a> <a id="4627" href="Categories.Yoneda.Properties.html#251" class="Bound">C</a> <a id="4629" href="Categories.Functor.Hom.html#3133" class="Function Operator">][-,</a> <a id="4634" href="Categories.Functor.Core.html#797" class="Function">G.₀</a> <a id="4638" href="Categories.Yoneda.Properties.html#4462" class="Bound">Y</a> <a id="4640" href="Categories.Functor.Hom.html#3133" class="Function Operator">]</a> <a id="4642" href="Agda.Builtin.Sigma.html#218" class="InductiveConstructor Operator">,</a> <a id="4644" class="Symbol">_}</a> <a id="4647" class="Symbol">(</a><a id="4648" href="Categories.Yoneda.Properties.html#1213" class="Function">idN</a> <a id="4652" href="Agda.Builtin.Sigma.html#218" class="InductiveConstructor Operator">,</a> <a id="4654" href="Categories.Functor.Core.html#806" class="Function">F.₁</a> <a id="4658" href="Categories.Yoneda.Properties.html#4465" class="Bound">f</a><a id="4659" class="Symbol">)</a> <a id="4661" class="Symbol">{</a><a id="4662" href="Categories.Yoneda.Properties.html#3902" class="Function">nat-appʳ</a> <a id="4671" href="Categories.Yoneda.Properties.html#4462" class="Bound">Y</a> <a id="4673" href="Categories.Yoneda.Properties.html#4368" class="Bound">α</a><a id="4674" class="Symbol">}</a> <a id="4676" class="Symbol">{</a><a id="4677" href="Categories.Yoneda.Properties.html#3902" class="Function">nat-appʳ</a> <a id="4686" href="Categories.Yoneda.Properties.html#4462" class="Bound">Y</a> <a id="4688" href="Categories.Yoneda.Properties.html#4368" class="Bound">α</a><a id="4689" class="Symbol">}</a> <a id="4691" class="Symbol">(</a><a id="4692" href="Function.Equality.html#1140" class="Field">cong</a> <a id="4697" class="Symbol">(</a><a id="4698" href="Categories.NaturalTransformation.Core.html#783" class="Field">η</a> <a id="4700" href="Categories.Yoneda.Properties.html#4368" class="Bound">α</a> <a id="4702" class="Symbol">_)))</a> <a id="4707" href="Relation.Binary.Reasoning.Setoid.html#1162" class="Function">⟩</a>
<a id="4719" href="Categories.NaturalTransformation.Core.html#783" class="Field">η</a> <a id="4721" href="Categories.Yoneda.Properties.html#4368" class="Bound">α</a> <a id="4723" class="Symbol">(</a><a id="4724" href="Categories.Functor.Core.html#797" class="Function">F.₀</a> <a id="4728" href="Categories.Yoneda.Properties.html#4460" class="Bound">X</a> <a id="4730" href="Agda.Builtin.Sigma.html#218" class="InductiveConstructor Operator">,</a> <a id="4732" href="Categories.Yoneda.Properties.html#4462" class="Bound">Y</a><a id="4733" class="Symbol">)</a> <a id="4735" href="Function.Equality.html#1073" class="Field Operator">⟨$⟩</a> <a id="4739" href="Categories.Functor.Core.html#806" class="Function">F.₁</a> <a id="4743" href="Categories.Yoneda.Properties.html#4465" class="Bound">f</a> <a id="4745" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="4747" href="Categories.Category.Core.html#630" class="Field">id</a> <a id="4758" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">≈⟨</a> <a id="4761" href="Function.Equality.html#1140" class="Field">cong</a> <a id="4766" class="Symbol">(</a><a id="4767" href="Categories.NaturalTransformation.Core.html#783" class="Field">η</a> <a id="4769" href="Categories.Yoneda.Properties.html#4368" class="Bound">α</a> <a id="4771" class="Symbol">(</a><a id="4772" href="Categories.Functor.Core.html#797" class="Function">F.₀</a> <a id="4776" href="Categories.Yoneda.Properties.html#4460" class="Bound">X</a> <a id="4778" href="Agda.Builtin.Sigma.html#218" class="InductiveConstructor Operator">,</a> <a id="4780" href="Categories.Yoneda.Properties.html#4462" class="Bound">Y</a><a id="4781" class="Symbol">))</a> <a id="4784" class="Symbol">(</a><a id="4785" href="Categories.Category.Core.html#1706" class="Function">∘-resp-≈ʳ</a> <a id="4795" class="Symbol">(</a><a id="4796" href="Categories.Category.Core.html#3005" class="Function">⟺</a> <a id="4798" href="Categories.Category.Core.html#1096" class="Field">identityˡ</a><a id="4807" class="Symbol">))</a> <a id="4810" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">⟩</a>
<a id="4822" href="Categories.NaturalTransformation.Core.html#783" class="Field">η</a> <a id="4824" href="Categories.Yoneda.Properties.html#4368" class="Bound">α</a> <a id="4826" class="Symbol">(</a><a id="4827" href="Categories.Functor.Core.html#797" class="Function">F.₀</a> <a id="4831" href="Categories.Yoneda.Properties.html#4460" class="Bound">X</a> <a id="4833" href="Agda.Builtin.Sigma.html#218" class="InductiveConstructor Operator">,</a> <a id="4835" href="Categories.Yoneda.Properties.html#4462" class="Bound">Y</a><a id="4836" class="Symbol">)</a> <a id="4838" href="Function.Equality.html#1073" class="Field Operator">⟨$⟩</a> <a id="4842" href="Categories.Functor.Core.html#806" class="Function">F.₁</a> <a id="4846" href="Categories.Yoneda.Properties.html#4465" class="Bound">f</a> <a id="4848" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="4850" href="Categories.Category.Core.html#630" class="Field">id</a> <a id="4853" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="4855" href="Categories.Category.Core.html#630" class="Field">id</a> <a id="4861" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">≈⟨</a> <a id="4864" href="Categories.NaturalTransformation.Core.html#827" class="Field">commute</a> <a id="4872" href="Categories.Yoneda.Properties.html#4368" class="Bound">α</a> <a id="4874" class="Symbol">(</a><a id="4875" href="Categories.Category.Core.html#630" class="Field">id</a> <a id="4878" href="Agda.Builtin.Sigma.html#218" class="InductiveConstructor Operator">,</a> <a id="4880" href="Categories.Yoneda.Properties.html#4465" class="Bound">f</a><a id="4881" class="Symbol">)</a> <a id="4883" href="Relation.Binary.Structures.html#1577" class="Function">CE.refl</a> <a id="4891" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">⟩</a>
<a id="4903" href="Categories.Functor.Core.html#806" class="Function">G.₁</a> <a id="4907" href="Categories.Yoneda.Properties.html#4465" class="Bound">f</a> <a id="4909" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="4911" class="Symbol">(</a><a id="4912" href="Categories.NaturalTransformation.Core.html#783" class="Field">η</a> <a id="4914" href="Categories.Yoneda.Properties.html#4368" class="Bound">α</a> <a id="4916" class="Symbol">(</a><a id="4917" href="Categories.Functor.Core.html#797" class="Function">F.₀</a> <a id="4921" href="Categories.Yoneda.Properties.html#4460" class="Bound">X</a> <a id="4923" href="Agda.Builtin.Sigma.html#218" class="InductiveConstructor Operator">,</a> <a id="4925" href="Categories.Yoneda.Properties.html#4460" class="Bound">X</a><a id="4926" class="Symbol">)</a> <a id="4928" href="Function.Equality.html#1073" class="Field Operator">⟨$⟩</a> <a id="4932" href="Categories.Category.Core.html#630" class="Field">id</a><a id="4934" class="Symbol">)</a> <a id="4936" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="4938" href="Categories.Category.Core.html#630" class="Field">id</a> <a id="4942" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">≈⟨</a> <a id="4945" href="Categories.Category.Core.html#2734" class="Function Operator">refl⟩∘⟨</a> <a id="4953" href="Categories.Category.Core.html#1145" class="Field">identityʳ</a> <a id="4963" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">⟩</a>
<a id="4975" href="Categories.Functor.Core.html#806" class="Function">G.₁</a> <a id="4979" href="Categories.Yoneda.Properties.html#4465" class="Bound">f</a> <a id="4981" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="4983" class="Symbol">(</a><a id="4984" href="Categories.NaturalTransformation.Core.html#783" class="Field">η</a> <a id="4986" href="Categories.Yoneda.Properties.html#4368" class="Bound">α</a> <a id="4988" class="Symbol">(</a><a id="4989" href="Categories.Functor.Core.html#797" class="Function">F.₀</a> <a id="4993" href="Categories.Yoneda.Properties.html#4460" class="Bound">X</a> <a id="4995" href="Agda.Builtin.Sigma.html#218" class="InductiveConstructor Operator">,</a> <a id="4997" href="Categories.Yoneda.Properties.html#4460" class="Bound">X</a><a id="4998" class="Symbol">)</a> <a id="5000" href="Function.Equality.html#1073" class="Field Operator">⟨$⟩</a> <a id="5004" href="Categories.Category.Core.html#630" class="Field">id</a><a id="5006" class="Symbol">)</a> <a id="5013" href="Relation.Binary.Reasoning.Base.Single.html#2564" class="Function Operator">∎</a>
<a id="5023" class="Symbol">}</a>
<a id="5028" class="Keyword">module</a> <a id="5035" href="Categories.Yoneda.Properties.html#5035" class="Module">_</a> <a id="5037" class="Symbol">(</a><a id="5038" href="Categories.Yoneda.Properties.html#5038" class="Bound">F</a> <a id="5040" href="Categories.Yoneda.Properties.html#5040" class="Bound">G</a> <a id="5042" class="Symbol">:</a> <a id="5044" href="Categories.Functor.Core.html#248" class="Record">Functor</a> <a id="5052" href="Categories.Yoneda.Properties.html#3464" class="Bound">D</a> <a id="5054" href="Categories.Yoneda.Properties.html#251" class="Bound">C</a><a id="5055" class="Symbol">)</a> <a id="5057" class="Keyword">where</a>
<a id="5067" class="Keyword">private</a>
<a id="5081" class="Keyword">module</a> <a id="5088" href="Categories.Yoneda.Properties.html#5088" class="Module">F</a> <a id="5090" class="Symbol">=</a> <a id="5092" href="Categories.Functor.Core.html#248" class="Module">Functor</a> <a id="5100" href="Categories.Yoneda.Properties.html#5038" class="Bound">F</a> <a id="5102" class="Keyword">using</a> <a id="5108" class="Symbol">(</a><a id="5109" href="Categories.Functor.Core.html#797" class="Function">₀</a><a id="5110" class="Symbol">)</a>
<a id="5118" class="Keyword">module</a> <a id="5125" href="Categories.Yoneda.Properties.html#5125" class="Module">G</a> <a id="5127" class="Symbol">=</a> <a id="5129" href="Categories.Functor.Core.html#248" class="Module">Functor</a> <a id="5137" href="Categories.Yoneda.Properties.html#5040" class="Bound">G</a> <a id="5139" class="Keyword">using</a> <a id="5145" class="Symbol">(</a><a id="5146" href="Categories.Functor.Core.html#797" class="Function">₀</a><a id="5147" class="Symbol">)</a>
<a id="5155" href="Categories.Yoneda.Properties.html#5155" class="Function">Hom[-,F-]</a> <a id="5165" class="Symbol">:</a> <a id="5167" href="Categories.Functor.Bifunctor.html#441" class="Function">Bifunctor</a> <a id="5177" href="Categories.Category.Core.html#3132" class="Function">C.op</a> <a id="5182" href="Categories.Yoneda.Properties.html#3464" class="Bound">D</a> <a id="5184" class="Symbol">(</a><a id="5185" href="Categories.Category.Instance.Setoids.html#316" class="Function">Setoids</a> <a id="5193" href="Categories.Yoneda.Properties.html#237" class="Bound">ℓ</a> <a id="5195" href="Categories.Yoneda.Properties.html#239" class="Bound">e</a><a id="5196" class="Symbol">)</a>
<a id="5204" href="Categories.Yoneda.Properties.html#5155" class="Function">Hom[-,F-]</a> <a id="5214" class="Symbol">=</a> <a id="5216" href="Categories.Functor.Hom.html#2973" class="Function Operator">Hom[</a> <a id="5221" href="Categories.Yoneda.Properties.html#251" class="Bound">C</a> <a id="5223" href="Categories.Functor.Hom.html#2973" class="Function Operator">][-,-]</a> <a id="5230" href="Categories.Functor.html#747" class="Function Operator">∘F</a> <a id="5233" class="Symbol">(</a><a id="5234" href="Categories.Yoneda.Properties.html#810" class="Function">idF</a> <a id="5238" href="Categories.Category.Product.html#1962" class="Function Operator">⁂</a> <a id="5240" href="Categories.Yoneda.Properties.html#5038" class="Bound">F</a><a id="5241" class="Symbol">)</a>
<a id="5250" href="Categories.Yoneda.Properties.html#5250" class="Function">Hom[-,G-]</a> <a id="5260" class="Symbol">:</a> <a id="5262" href="Categories.Functor.Bifunctor.html#441" class="Function">Bifunctor</a> <a id="5272" href="Categories.Category.Core.html#3132" class="Function">C.op</a> <a id="5277" href="Categories.Yoneda.Properties.html#3464" class="Bound">D</a> <a id="5279" class="Symbol">(</a><a id="5280" href="Categories.Category.Instance.Setoids.html#316" class="Function">Setoids</a> <a id="5288" href="Categories.Yoneda.Properties.html#237" class="Bound">ℓ</a> <a id="5290" href="Categories.Yoneda.Properties.html#239" class="Bound">e</a><a id="5291" class="Symbol">)</a>
<a id="5299" href="Categories.Yoneda.Properties.html#5250" class="Function">Hom[-,G-]</a> <a id="5309" class="Symbol">=</a> <a id="5311" href="Categories.Functor.Hom.html#2973" class="Function Operator">Hom[</a> <a id="5316" href="Categories.Yoneda.Properties.html#251" class="Bound">C</a> <a id="5318" href="Categories.Functor.Hom.html#2973" class="Function Operator">][-,-]</a> <a id="5325" href="Categories.Functor.html#747" class="Function Operator">∘F</a> <a id="5328" class="Symbol">(</a><a id="5329" href="Categories.Yoneda.Properties.html#810" class="Function">idF</a> <a id="5333" href="Categories.Category.Product.html#1962" class="Function Operator">⁂</a> <a id="5335" href="Categories.Yoneda.Properties.html#5040" class="Bound">G</a><a id="5336" class="Symbol">)</a>
<a id="5343" class="Comment">-- The implicits given below are sometimes needed (yellow), sometimes make an enormous difference in</a>
<a id="5448" class="Comment">-- typechecking time. For example, in yoneda.⇒.commute, &quot;nat-appʳ X F⇐G&quot; is needed, while</a>
<a id="5542" class="Comment">-- nat-appʳ {F = G} {F} X F⇐G makes a huge difference in checking time.</a>
<a id="5618" href="Categories.Yoneda.Properties.html#5618" class="Function">yoneda-NI</a> <a id="5628" class="Symbol">:</a> <a id="5630" href="Categories.NaturalTransformation.NaturalIsomorphism.html#651" class="Record">NaturalIsomorphism</a> <a id="5649" href="Categories.Yoneda.Properties.html#5155" class="Function">Hom[-,F-]</a> <a id="5659" href="Categories.Yoneda.Properties.html#5250" class="Function">Hom[-,G-]</a> <a id="5669" class="Symbol">→</a> <a id="5671" href="Categories.NaturalTransformation.NaturalIsomorphism.html#651" class="Record">NaturalIsomorphism</a> <a id="5690" href="Categories.Yoneda.Properties.html#5038" class="Bound">F</a> <a id="5692" href="Categories.Yoneda.Properties.html#5040" class="Bound">G</a>
<a id="5698" href="Categories.Yoneda.Properties.html#5618" class="Function">yoneda-NI</a> <a id="5708" href="Categories.Yoneda.Properties.html#5708" class="Bound">ni</a> <a id="5711" class="Symbol">=</a> <a id="5713" class="Keyword">record</a>
<a id="5726" class="Symbol">{</a> <a id="5728" href="Categories.NaturalTransformation.NaturalIsomorphism.html#891" class="Field">F⇒G</a> <a id="5732" class="Symbol">=</a> <a id="5734" href="Categories.Yoneda.Properties.html#4270" class="Function">transform</a> <a id="5744" href="Categories.NaturalTransformation.NaturalIsomorphism.html#891" class="Field">F⇒G</a>
<a id="5754" class="Symbol">;</a> <a id="5756" href="Categories.NaturalTransformation.NaturalIsomorphism.html#927" class="Field">F⇐G</a> <a id="5760" class="Symbol">=</a> <a id="5762" href="Categories.Yoneda.Properties.html#4270" class="Function">transform</a> <a id="5772" href="Categories.NaturalTransformation.NaturalIsomorphism.html#927" class="Field">F⇐G</a>
<a id="5782" class="Symbol">;</a> <a id="5784" href="Categories.NaturalTransformation.NaturalIsomorphism.html#1051" class="Field">iso</a> <a id="5788" class="Symbol">=</a> <a id="5790" class="Symbol">λ</a> <a id="5792" href="Categories.Yoneda.Properties.html#5792" class="Bound">X</a> <a id="5794" class="Symbol">→</a> <a id="5796" class="Keyword">record</a>
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<a id="7523" href="Categories.NaturalTransformation.Core.html#783" class="Function">⇒.η</a> <a id="7527" class="Symbol">(</a><a id="7528" href="Categories.Functor.Core.html#797" class="Function">G.₀</a> <a id="7532" href="Categories.Yoneda.Properties.html#5792" class="Bound">X</a> <a id="7534" href="Agda.Builtin.Sigma.html#218" class="InductiveConstructor Operator">,</a> <a id="7536" href="Categories.Yoneda.Properties.html#5792" class="Bound">X</a><a id="7537" class="Symbol">)</a> <a id="7539" href="Function.Equality.html#1073" class="Field Operator">⟨$⟩</a> <a id="7543" class="Symbol">(</a><a id="7544" href="Categories.NaturalTransformation.Core.html#783" class="Function">⇐.η</a> <a id="7548" class="Symbol">(</a><a id="7549" href="Categories.Functor.Core.html#797" class="Function">G.₀</a> <a id="7553" href="Categories.Yoneda.Properties.html#5792" class="Bound">X</a> <a id="7555" href="Agda.Builtin.Sigma.html#218" class="InductiveConstructor Operator">,</a> <a id="7557" href="Categories.Yoneda.Properties.html#5792" class="Bound">X</a><a id="7558" class="Symbol">)</a> <a id="7560" href="Function.Equality.html#1073" class="Field Operator">⟨$⟩</a> <a id="7564" href="Categories.Category.Core.html#630" class="Field">id</a><a id="7566" class="Symbol">)</a> <a id="7580" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">≈⟨</a> <a id="7583" href="Categories.Morphism.html#1612" class="Function">iso.isoʳ</a> <a id="7592" class="Symbol">_</a> <a id="7594" href="Relation.Binary.Structures.html#1577" class="Function">CE.refl</a> <a id="7602" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">⟩</a>
<a id="7614" href="Categories.Category.Core.html#630" class="Field">id</a> <a id="7670" href="Relation.Binary.Reasoning.Base.Single.html#2564" class="Function Operator">∎</a>
<a id="7680" class="Symbol">}</a>
<a id="7688" class="Symbol">}</a>
<a id="7696" class="Keyword">where</a>
<a id="7708" class="Keyword">open</a> <a id="7713" href="Categories.NaturalTransformation.NaturalIsomorphism.html#651" class="Module">NaturalIsomorphism</a> <a id="7732" href="Categories.Yoneda.Properties.html#5708" class="Bound">ni</a> <a id="7735" class="Keyword">using</a> <a id="7741" class="Symbol">(</a><a id="7742" href="Categories.NaturalTransformation.NaturalIsomorphism.html#891" class="Field">F⇒G</a><a id="7745" class="Symbol">;</a> <a id="7747" href="Categories.NaturalTransformation.NaturalIsomorphism.html#927" class="Field">F⇐G</a><a id="7750" class="Symbol">;</a> <a id="7752" class="Keyword">module</a> <a id="7759" href="Categories.NaturalTransformation.NaturalIsomorphism.html#1008" class="Module">⇐</a><a id="7760" class="Symbol">;</a> <a id="7762" class="Keyword">module</a> <a id="7769" href="Categories.NaturalTransformation.NaturalIsomorphism.html#969" class="Module">⇒</a><a id="7770" class="Symbol">;</a> <a id="7772" class="Keyword">module</a> <a id="7779" href="Categories.NaturalTransformation.NaturalIsomorphism.html#1104" class="Module">iso</a><a id="7782" class="Symbol">)</a>
<a id="7790" class="Keyword">open</a> <a id="7795" href="Categories.Morphism.Reasoning.html" class="Module">MR</a> <a id="7798" href="Categories.Yoneda.Properties.html#251" class="Bound">C</a> <a id="7800" class="Keyword">using</a> <a id="7806" class="Symbol">(</a><a id="7807" href="Categories.Morphism.Reasoning.Core.html#3063" class="Function">introˡ</a><a id="7813" class="Symbol">)</a>
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