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bsc-leon-vatthauer/agda/bsc-thesis/Categories.Yoneda.Properties.html
Leon Vatthauer 2bf4655eae
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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#742" 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>
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<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#2043" class="Record">Func</a><a id="356" class="Symbol">;</a> <a id="358" href="Function.Bundles.html#7340" class="Record">Inverse</a><a id="365" class="Symbol">;</a> <a id="367" href="Function.Bundles.html#15133" class="Function Operator">_⟨$⟩_</a><a id="372" class="Symbol">)</a>
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<a id="432" class="Keyword">import</a> <a id="439" href="Relation.Binary.Reasoning.Setoid.html" class="Module">Relation.Binary.Reasoning.Setoid</a> <a id="472" class="Symbol">as</a> <a id="475" class="Module">SetoidR</a>
<a id="483" class="Keyword">open</a> <a id="488" class="Keyword">import</a> <a id="495" href="Data.Product.html" class="Module">Data.Product</a> <a id="508" class="Keyword">using</a> <a id="514" class="Symbol">(</a><a id="515" href="Agda.Builtin.Sigma.html#235" class="InductiveConstructor Operator">_,_</a><a id="518" class="Symbol">;</a> <a id="520" href="Agda.Builtin.Sigma.html#165" class="Record">Σ</a><a id="521" class="Symbol">)</a>
<a id="524" class="Keyword">open</a> <a id="529" class="Keyword">import</a> <a id="536" href="Categories.Category.Product.html" class="Module">Categories.Category.Product</a> <a id="564" class="Keyword">using</a> <a id="570" class="Symbol">(</a><a id="571" href="Categories.Category.Product.html#1962" class="Function Operator">_⁂_</a><a id="574" class="Symbol">)</a>
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<a id="715" class="Keyword">open</a> <a id="720" class="Keyword">import</a> <a id="727" href="Categories.Functor.Properties.html" class="Module">Categories.Functor.Properties</a> <a id="757" class="Keyword">using</a> <a id="763" class="Symbol">(</a><a id="764" href="Categories.Functor.Properties.html#1132" class="Function">Full</a><a id="768" class="Symbol">;</a> <a id="770" href="Categories.Functor.Properties.html#987" class="Function">Faithful</a><a id="778" class="Symbol">;</a> <a id="780" href="Categories.Functor.Properties.html#1268" class="Function">FullyFaithful</a><a id="793" class="Symbol">)</a>
<a id="795" class="Keyword">open</a> <a id="800" class="Keyword">import</a> <a id="807" href="Categories.Functor.Hom.html" class="Module">Categories.Functor.Hom</a> <a id="830" class="Keyword">using</a> <a id="836" class="Symbol">(</a><a id="837" class="Keyword">module</a> <a id="844" href="Categories.Functor.Hom.html#625" class="Module">Hom</a><a id="847" class="Symbol">;</a> <a id="849" href="Categories.Functor.Hom.html#1738" class="Function Operator">Hom[_][-,_]</a><a id="860" class="Symbol">;</a> <a id="862" href="Categories.Functor.Hom.html#1578" class="Function Operator">Hom[_][-,-]</a><a id="873" class="Symbol">)</a>
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<a id="934" class="Keyword">open</a> <a id="939" class="Keyword">import</a> <a id="946" href="Categories.NaturalTransformation.html" class="Module">Categories.NaturalTransformation</a> <a id="979" class="Keyword">using</a> <a id="985" class="Symbol">(</a><a id="986" href="Categories.NaturalTransformation.Core.html#466" class="Record">NaturalTransformation</a><a id="1007" class="Symbol">;</a> <a id="1009" href="Categories.NaturalTransformation.Core.html#1750" class="Function">ntHelper</a><a id="1017" class="Symbol">)</a> <a id="1019" class="Keyword">renaming</a> <a id="1028" class="Symbol">(</a><a id="1029" href="Categories.NaturalTransformation.Core.html#2132" class="Function">id</a> <a id="1032" class="Symbol">to</a> <a id="1035" class="Function">idN</a><a id="1038" class="Symbol">)</a>
<a id="1040" class="Keyword">open</a> <a id="1045" class="Keyword">import</a> <a id="1052" href="Categories.NaturalTransformation.NaturalIsomorphism.html" class="Module">Categories.NaturalTransformation.NaturalIsomorphism</a> <a id="1104" class="Keyword">using</a> <a id="1110" class="Symbol">(</a><a id="1111" href="Categories.NaturalTransformation.NaturalIsomorphism.html#651" class="Record">NaturalIsomorphism</a><a id="1129" class="Symbol">)</a>
<a id="1131" class="Keyword">open</a> <a id="1136" class="Keyword">import</a> <a id="1143" href="Categories.Yoneda.html" class="Module">Categories.Yoneda</a> <a id="1161" class="Keyword">using</a> <a id="1167" class="Symbol">(</a><a id="1168" class="Keyword">module</a> <a id="1175" href="Categories.Yoneda.html#2049" class="Module">Yoneda</a><a id="1181" class="Symbol">)</a>
<a id="1184" class="Keyword">import</a> <a id="1191" href="Categories.Morphism.html" class="Module">Categories.Morphism</a> <a id="1211" class="Symbol">as</a> <a id="1214" class="Module">Mor</a>
<a id="1218" class="Keyword">import</a> <a id="1225" href="Categories.Morphism.Reasoning.html" class="Module">Categories.Morphism.Reasoning</a> <a id="1255" class="Symbol">as</a> <a id="1258" class="Module">MR</a>
<a id="1261" class="Keyword">import</a> <a id="1268" href="Categories.NaturalTransformation.Hom.html" class="Module">Categories.NaturalTransformation.Hom</a> <a id="1305" class="Symbol">as</a> <a id="1308" class="Module">NT-Hom</a>
<a id="1316" class="Keyword">open</a> <a id="1321" href="Categories.Category.Core.html#442" class="Module">Category</a> <a id="1330" href="Categories.Yoneda.Properties.html#251" class="Bound">C</a> <a id="1332" class="Keyword">using</a> <a id="1338" class="Symbol">(</a><a id="1339" class="Keyword">module</a> <a id="1346" href="Categories.Category.Core.html#2462" class="Module">HomReasoning</a><a id="1358" class="Symbol">;</a> <a id="1360" href="Categories.Category.Core.html#630" class="Field">id</a><a id="1362" class="Symbol">;</a> <a id="1364" href="Categories.Category.Core.html#656" class="Field Operator">_∘_</a><a id="1367" class="Symbol">;</a> <a id="1369" href="Categories.Category.Core.html#1096" class="Field">identityˡ</a><a id="1378" class="Symbol">;</a> <a id="1380" href="Categories.Category.Core.html#1145" class="Field">identityʳ</a><a id="1389" class="Symbol">;</a> <a id="1391" href="Categories.Category.Core.html#559" class="Field">Obj</a><a id="1394" class="Symbol">;</a> <a id="1396" href="Categories.Category.Core.html#1597" class="Function">∘-resp-≈ˡ</a><a id="1405" class="Symbol">;</a> <a id="1407" href="Categories.Category.Core.html#1706" class="Function">∘-resp-≈ʳ</a><a id="1416" class="Symbol">)</a>
<a id="1418" class="Keyword">open</a> <a id="1423" href="Categories.Category.Core.html#2462" class="Module">HomReasoning</a>
<a id="1436" class="Keyword">open</a> <a id="1441" href="Categories.NaturalTransformation.Core.html#466" class="Module">NaturalTransformation</a>
<a id="1463" class="Keyword">open</a> <a id="1468" href="Categories.Yoneda.html#2049" class="Module">Yoneda</a> <a id="1475" href="Categories.Yoneda.Properties.html#251" class="Bound">C</a> <a id="1477" class="Keyword">using</a> <a id="1483" class="Symbol">(</a><a id="1484" href="Categories.Yoneda.html#2422" class="Function">embed</a><a id="1489" class="Symbol">;</a> <a id="1491" href="Categories.Yoneda.html#2791" class="Function">yoneda-inverse</a><a id="1505" class="Symbol">;</a> <a id="1507" class="Keyword">module</a> <a id="1514" href="Categories.Yoneda.html#7311" class="Module">yoneda</a><a id="1520" class="Symbol">)</a>
<a id="1522" class="Keyword">private</a>
<a id="1532" class="Keyword">module</a> <a id="CE"></a><a id="1539" href="Categories.Yoneda.Properties.html#1539" class="Module">CE</a> <a id="1542" class="Symbol">=</a> <a id="1544" href="Categories.Category.Core.html#1530" class="Module">Category.Equiv</a> <a id="1559" href="Categories.Yoneda.Properties.html#251" class="Bound">C</a> <a id="1561" class="Keyword">using</a> <a id="1567" class="Symbol">(</a><a id="1568" href="Relation.Binary.Structures.html#1596" class="Function">refl</a><a id="1572" class="Symbol">)</a>
<a id="1576" class="Keyword">module</a> <a id="C"></a><a id="1583" href="Categories.Yoneda.Properties.html#1583" class="Module">C</a> <a id="1585" class="Symbol">=</a> <a id="1587" href="Categories.Category.Core.html#442" class="Module">Category</a> <a id="1596" href="Categories.Yoneda.Properties.html#251" class="Bound">C</a> <a id="1598" class="Keyword">using</a> <a id="1604" class="Symbol">(</a><a id="1605" href="Categories.Category.Core.html#3132" class="Function">op</a><a id="1607" class="Symbol">;</a> <a id="1609" href="Categories.Category.Core.html#630" class="Field">id</a><a id="1611" class="Symbol">)</a>
<a id="YoFull"></a><a id="1614" href="Categories.Yoneda.Properties.html#1614" class="Function">YoFull</a> <a id="1621" class="Symbol">:</a> <a id="1623" href="Categories.Functor.Properties.html#1132" class="Function">Full</a> <a id="1628" href="Categories.Yoneda.html#2422" class="Function">embed</a>
<a id="1634" href="Categories.Yoneda.Properties.html#1614" class="Function">YoFull</a> <a id="1641" class="Symbol">{</a><a id="1642" href="Categories.Yoneda.Properties.html#1642" class="Bound">X</a><a id="1643" class="Symbol">}</a> <a id="1645" class="Symbol">{</a><a id="1646" href="Categories.Yoneda.Properties.html#1646" class="Bound">Y</a><a id="1647" class="Symbol">}</a> <a id="1649" href="Categories.Yoneda.Properties.html#1649" class="Bound">ε</a> <a id="1651" class="Symbol">=</a> <a id="1653" href="Function.Bundles.html#2094" class="Field">Func.to</a> <a id="1661" class="Symbol">(</a><a id="1662" href="Categories.NaturalTransformation.Core.html#783" class="Field">η</a> <a id="1664" href="Categories.Yoneda.Properties.html#1649" class="Bound">ε</a> <a id="1666" href="Categories.Yoneda.Properties.html#1642" class="Bound">X</a><a id="1667" class="Symbol">)</a> <a id="1669" href="Categories.Category.Core.html#630" class="Field">id</a> <a id="1672" href="Agda.Builtin.Sigma.html#235" class="InductiveConstructor Operator">,</a> <a id="1674" class="Symbol">λ</a> <a id="1676" class="Symbol">{</a><a id="1677" href="Categories.Yoneda.Properties.html#1677" class="Bound">A</a><a id="1678" class="Symbol">}</a> <a id="1680" class="Symbol">{</a><a id="1681" href="Categories.Yoneda.Properties.html#1681" class="Bound">x</a><a id="1682" class="Symbol">}</a> <a id="1684" class="Symbol">→</a> <a id="1686" href="Relation.Binary.Reasoning.Syntax.html#1510" class="Function Operator">begin</a>
<a id="1694" href="Function.Bundles.html#2094" class="Field">Func.to</a> <a id="1702" class="Symbol">(</a><a id="1703" href="Categories.NaturalTransformation.Core.html#783" class="Field">η</a> <a id="1705" href="Categories.Yoneda.Properties.html#1649" class="Bound">ε</a> <a id="1707" href="Categories.Yoneda.Properties.html#1642" class="Bound">X</a><a id="1708" class="Symbol">)</a> <a id="1710" href="Categories.Category.Core.html#630" class="Field">id</a> <a id="1713" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="1715" href="Categories.Yoneda.Properties.html#1681" class="Bound">x</a> <a id="1724" href="Relation.Binary.Reasoning.Syntax.html#7400" class="Function">≈˘⟨</a> <a id="1728" href="Categories.Category.Core.html#1096" class="Field">identityˡ</a> <a id="1738" href="Relation.Binary.Reasoning.Syntax.html#7400" class="Function">⟩</a>
<a id="1742" href="Categories.Category.Core.html#630" class="Field">id</a> <a id="1745" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="1747" href="Function.Bundles.html#2094" class="Field">Func.to</a> <a id="1755" class="Symbol">(</a><a id="1756" href="Categories.NaturalTransformation.Core.html#783" class="Field">η</a> <a id="1758" href="Categories.Yoneda.Properties.html#1649" class="Bound">ε</a> <a id="1760" href="Categories.Yoneda.Properties.html#1642" class="Bound">X</a><a id="1761" class="Symbol">)</a> <a id="1763" href="Categories.Category.Core.html#630" class="Field">id</a> <a id="1766" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="1768" href="Categories.Yoneda.Properties.html#1681" class="Bound">x</a> <a id="1772" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="1775" href="Categories.NaturalTransformation.Core.html#1043" class="Field">sym-commute</a> <a id="1787" href="Categories.Yoneda.Properties.html#1649" class="Bound">ε</a> <a id="1789" href="Categories.Yoneda.Properties.html#1681" class="Bound">x</a> <a id="1791" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
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<a id="1874" href="Function.Bundles.html#2094" class="Field">Func.to</a> <a id="1882" class="Symbol">(</a><a id="1883" href="Categories.NaturalTransformation.Core.html#783" class="Field">η</a> <a id="1885" href="Categories.Yoneda.Properties.html#1649" class="Bound">ε</a> <a id="1887" href="Categories.Yoneda.Properties.html#1677" class="Bound">A</a><a id="1888" class="Symbol">)</a> <a id="1890" href="Categories.Yoneda.Properties.html#1681" class="Bound">x</a> <a id="1904" href="Relation.Binary.Reasoning.Syntax.html#12283" class="Function Operator">∎</a>
<a id="YoFaithful"></a><a id="1907" href="Categories.Yoneda.Properties.html#1907" class="Function">YoFaithful</a> <a id="1918" class="Symbol">:</a> <a id="1920" href="Categories.Functor.Properties.html#987" class="Function">Faithful</a> <a id="1929" href="Categories.Yoneda.html#2422" class="Function">embed</a>
<a id="1935" href="Categories.Yoneda.Properties.html#1907" class="Function">YoFaithful</a> <a id="1946" class="Symbol">{</a><a id="1947" href="Categories.Yoneda.Properties.html#1947" class="Bound">X</a><a id="1948" class="Symbol">}</a> <a id="1950" class="Symbol">{</a><a id="1951" href="Categories.Yoneda.Properties.html#1951" class="Bound">Y</a><a id="1952" class="Symbol">}</a> <a id="1954" class="Symbol">{</a><a id="1955" href="Categories.Yoneda.Properties.html#1955" class="Bound">f</a><a id="1956" class="Symbol">}</a> <a id="1958" class="Symbol">{</a><a id="1959" href="Categories.Yoneda.Properties.html#1959" class="Bound">g</a><a id="1960" class="Symbol">}</a> <a id="1962" href="Categories.Yoneda.Properties.html#1962" class="Bound">p</a> <a id="1964" class="Symbol">=</a> <a id="1966" href="Relation.Binary.Reasoning.Syntax.html#1510" class="Function Operator">begin</a>
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<a id="4446" href="Categories.NaturalTransformation.Core.html#783" class="Field">η</a> <a id="4448" href="Categories.Yoneda.Properties.html#4019" class="Bound">α</a> <a id="4450" class="Symbol">(</a><a id="4451" href="Categories.Functor.Core.html#797" class="Function">F.₀</a> <a id="4455" href="Categories.Yoneda.Properties.html#4111" class="Bound">X</a> <a id="4457" href="Agda.Builtin.Sigma.html#235" class="InductiveConstructor Operator">,</a> <a id="4459" href="Categories.Yoneda.Properties.html#4113" class="Bound">Y</a><a id="4460" class="Symbol">)</a> <a id="4462" href="Function.Bundles.html#15133" class="Function Operator">⟨$⟩</a> <a id="4466" href="Categories.Functor.Core.html#806" class="Function">F.₁</a> <a id="4470" href="Categories.Yoneda.Properties.html#4116" class="Bound">f</a> <a id="4472" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="4474" href="Categories.Category.Core.html#630" class="Field">id</a> <a id="4477" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="4479" href="Categories.Category.Core.html#630" class="Field">id</a> <a id="4484" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="4487" href="Categories.NaturalTransformation.Core.html#827" class="Field">commute</a> <a id="4495" href="Categories.Yoneda.Properties.html#4019" class="Bound">α</a> <a id="4497" class="Symbol">(</a><a id="4498" href="Categories.Category.Core.html#630" class="Field">id</a> <a id="4501" href="Agda.Builtin.Sigma.html#235" class="InductiveConstructor Operator">,</a> <a id="4503" href="Categories.Yoneda.Properties.html#4116" class="Bound">f</a><a id="4504" class="Symbol">)</a> <a id="4506" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
<a id="4518" href="Categories.Functor.Core.html#806" class="Function">G.₁</a> <a id="4522" href="Categories.Yoneda.Properties.html#4116" class="Bound">f</a> <a id="4524" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="4526" class="Symbol">(</a><a id="4527" href="Categories.NaturalTransformation.Core.html#783" class="Field">η</a> <a id="4529" href="Categories.Yoneda.Properties.html#4019" class="Bound">α</a> <a id="4531" class="Symbol">(</a><a id="4532" href="Categories.Functor.Core.html#797" class="Function">F.₀</a> <a id="4536" href="Categories.Yoneda.Properties.html#4111" class="Bound">X</a> <a id="4538" href="Agda.Builtin.Sigma.html#235" class="InductiveConstructor Operator">,</a> <a id="4540" href="Categories.Yoneda.Properties.html#4111" class="Bound">X</a><a id="4541" class="Symbol">)</a> <a id="4543" href="Function.Bundles.html#15133" class="Function Operator">⟨$⟩</a> <a id="4547" href="Categories.Category.Core.html#630" class="Field">id</a><a id="4549" class="Symbol">)</a> <a id="4551" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="4553" href="Categories.Category.Core.html#630" class="Field">id</a> <a id="4556" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="4559" href="Categories.Category.Core.html#2734" class="Function Operator">refl⟩∘⟨</a> <a id="4567" href="Categories.Category.Core.html#1145" class="Field">identityʳ</a> <a id="4577" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
<a id="4589" href="Categories.Functor.Core.html#806" class="Function">G.₁</a> <a id="4593" href="Categories.Yoneda.Properties.html#4116" class="Bound">f</a> <a id="4595" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="4597" class="Symbol">(</a><a id="4598" href="Categories.NaturalTransformation.Core.html#783" class="Field">η</a> <a id="4600" href="Categories.Yoneda.Properties.html#4019" class="Bound">α</a> <a id="4602" class="Symbol">(</a><a id="4603" href="Categories.Functor.Core.html#797" class="Function">F.₀</a> <a id="4607" href="Categories.Yoneda.Properties.html#4111" class="Bound">X</a> <a id="4609" href="Agda.Builtin.Sigma.html#235" class="InductiveConstructor Operator">,</a> <a id="4611" href="Categories.Yoneda.Properties.html#4111" class="Bound">X</a><a id="4612" class="Symbol">)</a> <a id="4614" href="Function.Bundles.html#15133" class="Function Operator">⟨$⟩</a> <a id="4618" href="Categories.Category.Core.html#630" class="Field">id</a><a id="4620" class="Symbol">)</a> <a id="4627" href="Relation.Binary.Reasoning.Syntax.html#12283" class="Function Operator">∎</a>
<a id="4637" class="Symbol">}</a>
<a id="4642" class="Keyword">module</a> <a id="4649" href="Categories.Yoneda.Properties.html#4649" class="Module">_</a> <a id="4651" class="Symbol">(</a><a id="4652" href="Categories.Yoneda.Properties.html#4652" class="Bound">F</a> <a id="4654" href="Categories.Yoneda.Properties.html#4654" class="Bound">G</a> <a id="4656" class="Symbol">:</a> <a id="4658" href="Categories.Functor.Core.html#248" class="Record">Functor</a> <a id="4666" href="Categories.Yoneda.Properties.html#3116" class="Bound">D</a> <a id="4668" href="Categories.Yoneda.Properties.html#251" class="Bound">C</a><a id="4669" class="Symbol">)</a> <a id="4671" class="Keyword">where</a>
<a id="4681" class="Keyword">private</a>
<a id="4695" class="Keyword">module</a> <a id="4702" href="Categories.Yoneda.Properties.html#4702" class="Module">F</a> <a id="4704" class="Symbol">=</a> <a id="4706" href="Categories.Functor.Core.html#248" class="Module">Functor</a> <a id="4714" href="Categories.Yoneda.Properties.html#4652" class="Bound">F</a> <a id="4716" class="Keyword">using</a> <a id="4722" class="Symbol">(</a><a id="4723" href="Categories.Functor.Core.html#797" class="Function">₀</a><a id="4724" class="Symbol">)</a>
<a id="4732" class="Keyword">module</a> <a id="4739" href="Categories.Yoneda.Properties.html#4739" class="Module">G</a> <a id="4741" class="Symbol">=</a> <a id="4743" href="Categories.Functor.Core.html#248" class="Module">Functor</a> <a id="4751" href="Categories.Yoneda.Properties.html#4654" class="Bound">G</a> <a id="4753" class="Keyword">using</a> <a id="4759" class="Symbol">(</a><a id="4760" href="Categories.Functor.Core.html#797" class="Function">₀</a><a id="4761" class="Symbol">)</a>
<a id="4769" href="Categories.Yoneda.Properties.html#4769" class="Function">Hom[-,F-]</a> <a id="4779" class="Symbol">:</a> <a id="4781" href="Categories.Functor.Bifunctor.html#441" class="Function">Bifunctor</a> <a id="4791" href="Categories.Category.Core.html#3132" class="Function">C.op</a> <a id="4796" href="Categories.Yoneda.Properties.html#3116" class="Bound">D</a> <a id="4798" class="Symbol">(</a><a id="4799" href="Categories.Category.Instance.Setoids.html#555" class="Function">Setoids</a> <a id="4807" href="Categories.Yoneda.Properties.html#237" class="Bound">ℓ</a> <a id="4809" href="Categories.Yoneda.Properties.html#239" class="Bound">e</a><a id="4810" class="Symbol">)</a>
<a id="4818" href="Categories.Yoneda.Properties.html#4769" class="Function">Hom[-,F-]</a> <a id="4828" class="Symbol">=</a> <a id="4830" href="Categories.Functor.Hom.html#1578" class="Function Operator">Hom[</a> <a id="4835" href="Categories.Yoneda.Properties.html#251" class="Bound">C</a> <a id="4837" href="Categories.Functor.Hom.html#1578" class="Function Operator">][-,-]</a> <a id="4844" href="Categories.Functor.html#747" class="Function Operator">∘F</a> <a id="4847" class="Symbol">(</a><a id="4848" href="Categories.Yoneda.Properties.html#710" class="Function">idF</a> <a id="4852" href="Categories.Category.Product.html#1962" class="Function Operator">⁂</a> <a id="4854" href="Categories.Yoneda.Properties.html#4652" class="Bound">F</a><a id="4855" class="Symbol">)</a>
<a id="4864" href="Categories.Yoneda.Properties.html#4864" class="Function">Hom[-,G-]</a> <a id="4874" class="Symbol">:</a> <a id="4876" href="Categories.Functor.Bifunctor.html#441" class="Function">Bifunctor</a> <a id="4886" href="Categories.Category.Core.html#3132" class="Function">C.op</a> <a id="4891" href="Categories.Yoneda.Properties.html#3116" class="Bound">D</a> <a id="4893" class="Symbol">(</a><a id="4894" href="Categories.Category.Instance.Setoids.html#555" class="Function">Setoids</a> <a id="4902" href="Categories.Yoneda.Properties.html#237" class="Bound">ℓ</a> <a id="4904" href="Categories.Yoneda.Properties.html#239" class="Bound">e</a><a id="4905" class="Symbol">)</a>
<a id="4913" href="Categories.Yoneda.Properties.html#4864" class="Function">Hom[-,G-]</a> <a id="4923" class="Symbol">=</a> <a id="4925" href="Categories.Functor.Hom.html#1578" class="Function Operator">Hom[</a> <a id="4930" href="Categories.Yoneda.Properties.html#251" class="Bound">C</a> <a id="4932" href="Categories.Functor.Hom.html#1578" class="Function Operator">][-,-]</a> <a id="4939" href="Categories.Functor.html#747" class="Function Operator">∘F</a> <a id="4942" class="Symbol">(</a><a id="4943" href="Categories.Yoneda.Properties.html#710" class="Function">idF</a> <a id="4947" href="Categories.Category.Product.html#1962" class="Function Operator">⁂</a> <a id="4949" href="Categories.Yoneda.Properties.html#4654" class="Bound">G</a><a id="4950" class="Symbol">)</a>
<a id="4957" class="Comment">-- The implicits given below are sometimes needed (yellow), sometimes make an enormous difference in</a>
<a id="5062" class="Comment">-- typechecking time. For example, in yoneda.⇒.commute, &quot;nat-appʳ X F⇐G&quot; is needed, while</a>
<a id="5156" class="Comment">-- nat-appʳ {F = G} {F} X F⇐G makes a huge difference in checking time.</a>
<a id="5232" href="Categories.Yoneda.Properties.html#5232" class="Function">yoneda-NI</a> <a id="5242" class="Symbol">:</a> <a id="5244" href="Categories.NaturalTransformation.NaturalIsomorphism.html#651" class="Record">NaturalIsomorphism</a> <a id="5263" href="Categories.Yoneda.Properties.html#4769" class="Function">Hom[-,F-]</a> <a id="5273" href="Categories.Yoneda.Properties.html#4864" class="Function">Hom[-,G-]</a> <a id="5283" class="Symbol">→</a> <a id="5285" href="Categories.NaturalTransformation.NaturalIsomorphism.html#651" class="Record">NaturalIsomorphism</a> <a id="5304" href="Categories.Yoneda.Properties.html#4652" class="Bound">F</a> <a id="5306" href="Categories.Yoneda.Properties.html#4654" class="Bound">G</a>
<a id="5312" href="Categories.Yoneda.Properties.html#5232" class="Function">yoneda-NI</a> <a id="5322" href="Categories.Yoneda.Properties.html#5322" class="Bound">ni</a> <a id="5325" class="Symbol">=</a> <a id="5327" class="Keyword">record</a>
<a id="5340" class="Symbol">{</a> <a id="5342" href="Categories.NaturalTransformation.NaturalIsomorphism.html#891" class="Field">F⇒G</a> <a id="5346" class="Symbol">=</a> <a id="5348" href="Categories.Yoneda.Properties.html#3921" class="Function">transform</a> <a id="5358" href="Categories.NaturalTransformation.NaturalIsomorphism.html#891" class="Field">F⇒G</a>
<a id="5368" class="Symbol">;</a> <a id="5370" href="Categories.NaturalTransformation.NaturalIsomorphism.html#927" class="Field">F⇐G</a> <a id="5374" class="Symbol">=</a> <a id="5376" href="Categories.Yoneda.Properties.html#3921" class="Function">transform</a> <a id="5386" href="Categories.NaturalTransformation.NaturalIsomorphism.html#927" class="Field">F⇐G</a>
<a id="5396" class="Symbol">;</a> <a id="5398" href="Categories.NaturalTransformation.NaturalIsomorphism.html#1051" class="Field">iso</a> <a id="5402" class="Symbol">=</a> <a id="5404" class="Symbol">λ</a> <a id="5406" href="Categories.Yoneda.Properties.html#5406" class="Bound">X</a> <a id="5408" class="Symbol">→</a> <a id="5410" class="Keyword">record</a>
<a id="5425" class="Symbol">{</a> <a id="5427" href="Categories.Morphism.html#1586" class="Field">isoˡ</a> <a id="5432" class="Symbol">=</a> <a id="5434" href="Relation.Binary.Reasoning.Syntax.html#1510" class="Function Operator">begin</a>
<a id="5450" class="Symbol">(</a><a id="5451" href="Categories.NaturalTransformation.Core.html#783" class="Function">⇐.η</a> <a id="5455" class="Symbol">(</a><a id="5456" href="Categories.Functor.Core.html#797" class="Function">G.₀</a> <a id="5460" href="Categories.Yoneda.Properties.html#5406" class="Bound">X</a> <a id="5462" href="Agda.Builtin.Sigma.html#235" class="InductiveConstructor Operator">,</a> <a id="5464" href="Categories.Yoneda.Properties.html#5406" class="Bound">X</a><a id="5465" class="Symbol">)</a> <a id="5467" href="Function.Bundles.html#15133" class="Function Operator">⟨$⟩</a> <a id="5471" href="Categories.Category.Core.html#630" class="Field">id</a><a id="5473" class="Symbol">)</a> <a id="5475" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="5477" class="Symbol">(</a><a id="5478" href="Categories.NaturalTransformation.Core.html#783" class="Function">⇒.η</a> <a id="5482" class="Symbol">(</a><a id="5483" href="Categories.Functor.Core.html#797" class="Function">F.₀</a> <a id="5487" href="Categories.Yoneda.Properties.html#5406" class="Bound">X</a> <a id="5489" href="Agda.Builtin.Sigma.html#235" class="InductiveConstructor Operator">,</a> <a id="5491" href="Categories.Yoneda.Properties.html#5406" class="Bound">X</a><a id="5492" class="Symbol">)</a> <a id="5494" href="Function.Bundles.html#15133" class="Function Operator">⟨$⟩</a> <a id="5498" href="Categories.Category.Core.html#630" class="Field">id</a><a id="5500" class="Symbol">)</a> <a id="5507" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="5510" href="Categories.Morphism.Reasoning.Core.html#3063" class="Function">introˡ</a> <a id="5517" href="Relation.Binary.Structures.html#1596" class="Function">CE.refl</a> <a id="5525" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
<a id="5537" href="Categories.Category.Core.html#630" class="Field">id</a> <a id="5540" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="5542" class="Symbol">(</a><a id="5543" href="Categories.NaturalTransformation.Core.html#783" class="Function">⇐.η</a> <a id="5547" class="Symbol">(</a><a id="5548" href="Categories.Functor.Core.html#797" class="Function">G.₀</a> <a id="5552" href="Categories.Yoneda.Properties.html#5406" class="Bound">X</a> <a id="5554" href="Agda.Builtin.Sigma.html#235" class="InductiveConstructor Operator">,</a> <a id="5556" href="Categories.Yoneda.Properties.html#5406" class="Bound">X</a><a id="5557" class="Symbol">)</a> <a id="5559" href="Function.Bundles.html#15133" class="Function Operator">⟨$⟩</a> <a id="5563" href="Categories.Category.Core.html#630" class="Field">id</a><a id="5565" class="Symbol">)</a> <a id="5567" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="5569" class="Symbol">(</a><a id="5570" href="Categories.NaturalTransformation.Core.html#783" class="Function">⇒.η</a> <a id="5574" class="Symbol">(</a><a id="5575" href="Categories.Functor.Core.html#797" class="Function">F.₀</a> <a id="5579" href="Categories.Yoneda.Properties.html#5406" class="Bound">X</a> <a id="5581" href="Agda.Builtin.Sigma.html#235" class="InductiveConstructor Operator">,</a> <a id="5583" href="Categories.Yoneda.Properties.html#5406" class="Bound">X</a><a id="5584" class="Symbol">)</a> <a id="5586" href="Function.Bundles.html#15133" class="Function Operator">⟨$⟩</a> <a id="5590" href="Categories.Category.Core.html#630" class="Field">id</a><a id="5592" class="Symbol">)</a> <a id="5594" href="Relation.Binary.Reasoning.Syntax.html#7400" class="Function">≈˘⟨</a> <a id="5598" href="Level.html#479" class="Field">lower</a> <a id="5604" class="Symbol">{</a><a id="5605" href="Categories.Yoneda.Properties.html#239" class="Bound">e</a><a id="5606" class="Symbol">}</a> <a id="5608" class="Symbol">(</a><a id="5609" href="Categories.NaturalTransformation.Core.html#827" class="Function">yoneda.⇒.commute</a> <a id="5626" class="Symbol">{</a><a id="5627" class="Argument">Y</a> <a id="5629" class="Symbol">=</a> <a id="5631" href="Categories.Functor.Hom.html#1738" class="Function Operator">Hom[</a> <a id="5636" href="Categories.Yoneda.Properties.html#251" class="Bound">C</a> <a id="5638" href="Categories.Functor.Hom.html#1738" class="Function Operator">][-,</a> <a id="5643" href="Categories.Functor.Core.html#797" class="Function">F.₀</a> <a id="5647" href="Categories.Yoneda.Properties.html#5406" class="Bound">X</a> <a id="5649" href="Categories.Functor.Hom.html#1738" class="Function Operator">]</a> <a id="5651" href="Agda.Builtin.Sigma.html#235" class="InductiveConstructor Operator">,</a> <a id="5653" class="Symbol">_}</a>
<a id="5738" class="Symbol">(</a><a id="5739" href="Categories.Yoneda.Properties.html#1035" class="Function">idN</a> <a id="5743" href="Agda.Builtin.Sigma.html#235" class="InductiveConstructor Operator">,</a> <a id="5745" class="Symbol">(</a><a id="5746" href="Categories.NaturalTransformation.Core.html#783" class="Function">⇒.η</a> <a id="5750" class="Symbol">(</a><a id="5751" href="Categories.Functor.Core.html#797" class="Function">F.₀</a> <a id="5755" href="Categories.Yoneda.Properties.html#5406" class="Bound">X</a> <a id="5757" href="Agda.Builtin.Sigma.html#235" class="InductiveConstructor Operator">,</a> <a id="5759" href="Categories.Yoneda.Properties.html#5406" class="Bound">X</a><a id="5760" class="Symbol">)</a> <a id="5762" href="Function.Bundles.html#15133" class="Function Operator">⟨$⟩</a> <a id="5766" href="Categories.Category.Core.html#630" class="Field">id</a><a id="5768" class="Symbol">))</a>
<a id="5853" class="Symbol">{</a><a id="5854" href="Categories.Yoneda.Properties.html#3554" class="Function">nat-appʳ</a> <a id="5863" class="Symbol">{</a><a id="5864" class="Argument">F</a> <a id="5866" class="Symbol">=</a> <a id="5868" href="Categories.Yoneda.Properties.html#4654" class="Bound">G</a><a id="5869" class="Symbol">}</a> <a id="5871" class="Symbol">{</a><a id="5872" href="Categories.Yoneda.Properties.html#4652" class="Bound">F</a><a id="5873" class="Symbol">}</a> <a id="5875" href="Categories.Yoneda.Properties.html#5406" class="Bound">X</a> <a id="5877" href="Categories.NaturalTransformation.NaturalIsomorphism.html#927" class="Field">F⇐G</a><a id="5880" class="Symbol">}</a> <a id="5882" class="Symbol">)</a> <a id="5885" href="Relation.Binary.Reasoning.Syntax.html#7400" class="Function">⟩</a>
<a id="5897" href="Categories.NaturalTransformation.Core.html#783" class="Function">⇐.η</a> <a id="5901" class="Symbol">(</a><a id="5902" href="Categories.Functor.Core.html#797" class="Function">F.₀</a> <a id="5906" href="Categories.Yoneda.Properties.html#5406" class="Bound">X</a> <a id="5908" href="Agda.Builtin.Sigma.html#235" class="InductiveConstructor Operator">,</a> <a id="5910" href="Categories.Yoneda.Properties.html#5406" class="Bound">X</a><a id="5911" class="Symbol">)</a> <a id="5913" href="Function.Bundles.html#15133" class="Function Operator">⟨$⟩</a> <a id="5917" class="Symbol">(</a><a id="5918" href="Categories.NaturalTransformation.Core.html#783" class="Function">⇒.η</a> <a id="5922" class="Symbol">(</a><a id="5923" href="Categories.Functor.Core.html#797" class="Function">F.₀</a> <a id="5927" href="Categories.Yoneda.Properties.html#5406" class="Bound">X</a> <a id="5929" href="Agda.Builtin.Sigma.html#235" class="InductiveConstructor Operator">,</a> <a id="5931" href="Categories.Yoneda.Properties.html#5406" class="Bound">X</a><a id="5932" class="Symbol">)</a> <a id="5934" href="Function.Bundles.html#15133" class="Function Operator">⟨$⟩</a> <a id="5938" href="Categories.Category.Core.html#630" class="Field">id</a><a id="5940" class="Symbol">)</a> <a id="5942" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="5944" href="Categories.Category.Core.html#630" class="Field">id</a> <a id="5954" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="5957" href="Function.Bundles.html#2113" class="Field">Func.cong</a> <a id="5967" class="Symbol">(</a><a id="5968" href="Categories.NaturalTransformation.Core.html#783" class="Function">⇐.η</a> <a id="5972" class="Symbol">_)</a> <a id="5975" href="Categories.Category.Core.html#1145" class="Field">identityʳ</a> <a id="5985" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
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