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<html><head><meta charset="utf-8"><title>Categories.Adjoint.TwoSided</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.Adjoint.TwoSided.html" class="Module">Categories.Adjoint.TwoSided</a> <a id="72" class="Keyword">where</a>
<a id="79" class="Comment">-- A &quot;two sided&quot; adjoint is an adjoint of two functors L and R where the</a>
<a id="152" class="Comment">-- unit and counit are natural isomorphisms.</a>
<a id="198" class="Comment">-- A two sided adjoint is the underlying data to an Adjoint Equivalence</a>
<a id="270" class="Keyword">open</a> <a id="275" class="Keyword">import</a> <a id="282" href="Level.html" class="Module">Level</a>
<a id="289" class="Keyword">open</a> <a id="294" class="Keyword">import</a> <a id="301" href="Categories.Adjoint.html" class="Module">Categories.Adjoint</a>
<a id="320" class="Keyword">open</a> <a id="325" class="Keyword">import</a> <a id="332" href="Categories.Category.Core.html" class="Module">Categories.Category.Core</a> <a id="357" class="Keyword">using</a> <a id="363" class="Symbol">(</a><a id="364" href="Categories.Category.Core.html#442" class="Record">Category</a><a id="372" class="Symbol">)</a>
<a id="374" class="Keyword">open</a> <a id="379" class="Keyword">import</a> <a id="386" href="Categories.Functor.html" class="Module">Categories.Functor</a> <a id="405" class="Keyword">renaming</a> <a id="414" class="Symbol">(</a><a id="415" href="Categories.Functor.html#349" class="Function">id</a> <a id="418" class="Symbol">to</a> <a id="421" class="Function">idF</a><a id="424" class="Symbol">)</a>
<a id="426" class="Keyword">open</a> <a id="431" class="Keyword">import</a> <a id="438" href="Categories.Functor.Properties.html" class="Module">Categories.Functor.Properties</a>
<a id="468" class="Keyword">open</a> <a id="473" class="Keyword">import</a> <a id="480" href="Categories.NaturalTransformation.html" class="Module">Categories.NaturalTransformation</a> <a id="513" class="Keyword">using</a> <a id="519" class="Symbol">(</a><a id="520" href="Categories.NaturalTransformation.Core.html#1750" class="Function">ntHelper</a><a id="528" class="Symbol">)</a>
<a id="530" class="Keyword">open</a> <a id="535" class="Keyword">import</a> <a id="542" href="Categories.NaturalTransformation.NaturalIsomorphism.html" class="Module">Categories.NaturalTransformation.NaturalIsomorphism</a> <a id="594" class="Symbol">as</a> <a id="597" class="Module">≃</a> <a id="599" class="Keyword">using</a> <a id="605" class="Symbol">(</a><a id="606" href="Categories.NaturalTransformation.NaturalIsomorphism.html#3600" class="Function Operator">_≃_</a><a id="609" class="Symbol">;</a> <a id="611" href="Categories.NaturalTransformation.NaturalIsomorphism.html#651" class="Record">NaturalIsomorphism</a><a id="629" class="Symbol">)</a>
<a id="631" class="Keyword">open</a> <a id="636" class="Keyword">import</a> <a id="643" href="Categories.NaturalTransformation.NaturalIsomorphism.Properties.html" class="Module">Categories.NaturalTransformation.NaturalIsomorphism.Properties</a>
<a id="706" class="Keyword">import</a> <a id="713" href="Categories.Morphism.Reasoning.html" class="Module">Categories.Morphism.Reasoning</a> <a id="743" class="Symbol">as</a> <a id="746" class="Module">MR</a>
<a id="750" class="Keyword">private</a>
<a id="760" class="Keyword">variable</a>
<a id="773" href="Categories.Adjoint.TwoSided.html#773" class="Generalizable">o</a> <a id="775" href="Categories.Adjoint.TwoSided.html#775" class="Generalizable">ℓ</a> <a id="777" href="Categories.Adjoint.TwoSided.html#777" class="Generalizable">e</a> <a id="782" class="Symbol">:</a> <a id="784" href="Agda.Primitive.html#591" class="Postulate">Level</a>
<a id="794" href="Categories.Adjoint.TwoSided.html#794" class="Generalizable">o′</a> <a id="797" href="Categories.Adjoint.TwoSided.html#797" class="Generalizable">ℓ′</a> <a id="800" href="Categories.Adjoint.TwoSided.html#800" class="Generalizable">e′</a> <a id="803" class="Symbol">:</a> <a id="805" href="Agda.Primitive.html#591" class="Postulate">Level</a>
<a id="815" href="Categories.Adjoint.TwoSided.html#815" class="Generalizable">C</a> <a id="817" href="Categories.Adjoint.TwoSided.html#817" class="Generalizable">D</a> <a id="824" class="Symbol">:</a> <a id="826" href="Categories.Category.Core.html#442" class="Record">Category</a> <a id="835" href="Categories.Adjoint.TwoSided.html#773" class="Generalizable">o</a> <a id="837" href="Categories.Adjoint.TwoSided.html#775" class="Generalizable">ℓ</a> <a id="839" href="Categories.Adjoint.TwoSided.html#777" class="Generalizable">e</a>
<a id="842" class="Keyword">infix</a> <a id="848" class="Number">5</a> <a id="850" href="Categories.Adjoint.TwoSided.html#863" class="Record Operator">_⊣⊢_</a>
<a id="856" class="Keyword">record</a> <a id="_⊣⊢_"></a><a id="863" href="Categories.Adjoint.TwoSided.html#863" class="Record Operator">_⊣⊢_</a> <a id="868" class="Symbol">(</a><a id="869" href="Categories.Adjoint.TwoSided.html#869" class="Bound">L</a> <a id="871" class="Symbol">:</a> <a id="873" href="Categories.Functor.Core.html#248" class="Record">Functor</a> <a id="881" href="Categories.Adjoint.TwoSided.html#815" class="Generalizable">C</a> <a id="883" href="Categories.Adjoint.TwoSided.html#817" class="Generalizable">D</a><a id="884" class="Symbol">)</a> <a id="886" class="Symbol">(</a><a id="887" href="Categories.Adjoint.TwoSided.html#887" class="Bound">R</a> <a id="889" class="Symbol">:</a> <a id="891" href="Categories.Functor.Core.html#248" class="Record">Functor</a> <a id="899" href="Categories.Adjoint.TwoSided.html#817" class="Generalizable">D</a> <a id="901" href="Categories.Adjoint.TwoSided.html#815" class="Generalizable">C</a><a id="902" class="Symbol">)</a> <a id="904" class="Symbol">:</a> <a id="906" href="Agda.Primitive.html#320" class="Primitive">Set</a> <a id="910" class="Symbol">(</a><a id="911" href="Level.html#602" class="Function">levelOfTerm</a> <a id="923" href="Categories.Adjoint.TwoSided.html#869" class="Bound">L</a> <a id="925" href="Agda.Primitive.html#804" class="Primitive Operator">⊔</a> <a id="927" href="Level.html#602" class="Function">levelOfTerm</a> <a id="939" href="Categories.Adjoint.TwoSided.html#887" class="Bound">R</a><a id="940" class="Symbol">)</a> <a id="942" class="Keyword">where</a>
<a id="950" class="Keyword">field</a>
<a id="_⊣⊢_.unit"></a><a id="960" href="Categories.Adjoint.TwoSided.html#960" class="Field">unit</a> <a id="967" class="Symbol">:</a> <a id="969" href="Categories.Adjoint.TwoSided.html#421" class="Function">idF</a> <a id="973" href="Categories.NaturalTransformation.NaturalIsomorphism.html#3600" class="Function Operator">≃</a> <a id="975" class="Symbol">(</a><a id="976" href="Categories.Adjoint.TwoSided.html#887" class="Bound">R</a> <a id="978" href="Categories.Functor.html#747" class="Function Operator">∘F</a> <a id="981" href="Categories.Adjoint.TwoSided.html#869" class="Bound">L</a><a id="982" class="Symbol">)</a>
<a id="_⊣⊢_.counit"></a><a id="988" href="Categories.Adjoint.TwoSided.html#988" class="Field">counit</a> <a id="995" class="Symbol">:</a> <a id="997" class="Symbol">(</a><a id="998" href="Categories.Adjoint.TwoSided.html#869" class="Bound">L</a> <a id="1000" href="Categories.Functor.html#747" class="Function Operator">∘F</a> <a id="1003" href="Categories.Adjoint.TwoSided.html#887" class="Bound">R</a><a id="1004" class="Symbol">)</a> <a id="1006" href="Categories.NaturalTransformation.NaturalIsomorphism.html#3600" class="Function Operator">≃</a> <a id="1008" href="Categories.Adjoint.TwoSided.html#421" class="Function">idF</a>
<a id="1015" class="Keyword">module</a> <a id="_⊣⊢_.unit"></a><a id="1022" href="Categories.Adjoint.TwoSided.html#1022" class="Module">unit</a> <a id="1029" class="Symbol">=</a> <a id="1031" href="Categories.NaturalTransformation.NaturalIsomorphism.html#651" class="Module">NaturalIsomorphism</a> <a id="1050" href="Categories.Adjoint.TwoSided.html#960" class="Field">unit</a>
<a id="1057" class="Keyword">module</a> <a id="_⊣⊢_.counit"></a><a id="1064" href="Categories.Adjoint.TwoSided.html#1064" class="Module">counit</a> <a id="1071" class="Symbol">=</a> <a id="1073" href="Categories.NaturalTransformation.NaturalIsomorphism.html#651" class="Module">NaturalIsomorphism</a> <a id="1092" href="Categories.Adjoint.TwoSided.html#988" class="Field">counit</a>
<a id="1102" class="Keyword">private</a>
<a id="1114" class="Keyword">module</a> <a id="_⊣⊢_.C"></a><a id="1121" href="Categories.Adjoint.TwoSided.html#1121" class="Module">C</a> <a id="1123" class="Symbol">=</a> <a id="1125" href="Categories.Category.Core.html#442" class="Module">Category</a> <a id="1134" href="Categories.Adjoint.TwoSided.html#881" class="Bound">C</a> <a id="1136" class="Keyword">using</a> <a id="1142" class="Symbol">(</a><a id="1143" href="Categories.Category.Core.html#559" class="Field">Obj</a><a id="1146" class="Symbol">;</a> <a id="1148" href="Categories.Category.Core.html#630" class="Field">id</a><a id="1150" class="Symbol">;</a> <a id="1152" href="Categories.Category.Core.html#656" class="Field Operator">_∘_</a><a id="1155" class="Symbol">;</a> <a id="1157" href="Categories.Category.Core.html#595" class="Field Operator">_≈_</a><a id="1160" class="Symbol">;</a> <a id="1162" class="Keyword">module</a> <a id="1169" href="Categories.Category.Core.html#2462" class="Module">HomReasoning</a><a id="1181" class="Symbol">)</a>
<a id="1187" class="Keyword">module</a> <a id="_⊣⊢_.D"></a><a id="1194" href="Categories.Adjoint.TwoSided.html#1194" class="Module">D</a> <a id="1196" class="Symbol">=</a> <a id="1198" href="Categories.Category.Core.html#442" class="Module">Category</a> <a id="1207" href="Categories.Adjoint.TwoSided.html#883" class="Bound">D</a> <a id="1209" class="Keyword">using</a> <a id="1215" class="Symbol">(</a><a id="1216" href="Categories.Category.Core.html#559" class="Field">Obj</a><a id="1219" class="Symbol">;</a> <a id="1221" href="Categories.Category.Core.html#630" class="Field">id</a><a id="1223" class="Symbol">;</a> <a id="1225" href="Categories.Category.Core.html#656" class="Field Operator">_∘_</a><a id="1228" class="Symbol">;</a> <a id="1230" href="Categories.Category.Core.html#595" class="Field Operator">_≈_</a><a id="1233" class="Symbol">;</a> <a id="1235" class="Keyword">module</a> <a id="1242" href="Categories.Category.Core.html#2462" class="Module">HomReasoning</a><a id="1254" class="Symbol">)</a>
<a id="1260" class="Keyword">module</a> <a id="_⊣⊢_.L"></a><a id="1267" href="Categories.Adjoint.TwoSided.html#1267" class="Module">L</a> <a id="1269" class="Symbol">=</a> <a id="1271" href="Categories.Functor.Core.html#248" class="Module">Functor</a> <a id="1279" href="Categories.Adjoint.TwoSided.html#869" class="Bound">L</a> <a id="1281" class="Keyword">using</a> <a id="1287" class="Symbol">(</a><a id="1288" href="Categories.Functor.Core.html#797" class="Function">₀</a><a id="1289" class="Symbol">;</a> <a id="1291" href="Categories.Functor.Core.html#806" class="Function">₁</a><a id="1292" class="Symbol">;</a> <a id="1294" href="Categories.Functor.Core.html#816" class="Function">op</a><a id="1296" class="Symbol">;</a> <a id="1298" href="Categories.Functor.Core.html#511" class="Field">identity</a><a id="1306" class="Symbol">)</a>
<a id="1312" class="Keyword">module</a> <a id="_⊣⊢_.R"></a><a id="1319" href="Categories.Adjoint.TwoSided.html#1319" class="Module">R</a> <a id="1321" class="Symbol">=</a> <a id="1323" href="Categories.Functor.Core.html#248" class="Module">Functor</a> <a id="1331" href="Categories.Adjoint.TwoSided.html#887" class="Bound">R</a> <a id="1333" class="Keyword">using</a> <a id="1339" class="Symbol">(</a><a id="1340" href="Categories.Functor.Core.html#797" class="Function">₀</a><a id="1341" class="Symbol">;</a> <a id="1343" href="Categories.Functor.Core.html#806" class="Function">₁</a><a id="1344" class="Symbol">;</a> <a id="1346" href="Categories.Functor.Core.html#816" class="Function">op</a><a id="1348" class="Symbol">;</a> <a id="1350" href="Categories.Functor.Core.html#511" class="Field">identity</a><a id="1358" class="Symbol">)</a>
<a id="1363" class="Keyword">field</a>
<a id="_⊣⊢_.zig"></a><a id="1373" href="Categories.Adjoint.TwoSided.html#1373" class="Field">zig</a> <a id="1377" class="Symbol">:</a> <a id="1379" class="Symbol">∀</a> <a id="1381" class="Symbol">{</a><a id="1382" href="Categories.Adjoint.TwoSided.html#1382" class="Bound">A</a> <a id="1384" class="Symbol">:</a> <a id="1386" href="Categories.Category.Core.html#559" class="Function">C.Obj</a><a id="1391" class="Symbol">}</a> <a id="1393" class="Symbol">→</a> <a id="1395" href="Categories.NaturalTransformation.Core.html#783" class="Function">counit.⇒.η</a> <a id="1406" class="Symbol">(</a><a id="1407" href="Categories.Functor.Core.html#797" class="Function">L.₀</a> <a id="1411" href="Categories.Adjoint.TwoSided.html#1382" class="Bound">A</a><a id="1412" class="Symbol">)</a> <a id="1414" href="Categories.Category.Core.html#656" class="Function Operator">D.∘</a> <a id="1418" href="Categories.Functor.Core.html#806" class="Function">L.₁</a> <a id="1422" class="Symbol">(</a><a id="1423" href="Categories.NaturalTransformation.Core.html#783" class="Function">unit.⇒.η</a> <a id="1432" href="Categories.Adjoint.TwoSided.html#1382" class="Bound">A</a><a id="1433" class="Symbol">)</a> <a id="1435" href="Categories.Category.Core.html#595" class="Function Operator">D.≈</a> <a id="1439" href="Categories.Category.Core.html#630" class="Function">D.id</a>
<a id="_⊣⊢_.zag"></a><a id="1448" href="Categories.Adjoint.TwoSided.html#1448" class="Field">zag</a> <a id="1452" class="Symbol">:</a> <a id="1454" class="Symbol">∀</a> <a id="1456" class="Symbol">{</a><a id="1457" href="Categories.Adjoint.TwoSided.html#1457" class="Bound">B</a> <a id="1459" class="Symbol">:</a> <a id="1461" href="Categories.Category.Core.html#559" class="Function">D.Obj</a><a id="1466" class="Symbol">}</a> <a id="1468" class="Symbol">→</a> <a id="1470" href="Categories.Functor.Core.html#806" class="Function">R.₁</a> <a id="1474" class="Symbol">(</a><a id="1475" href="Categories.NaturalTransformation.Core.html#783" class="Function">counit.⇒.η</a> <a id="1486" href="Categories.Adjoint.TwoSided.html#1457" class="Bound">B</a><a id="1487" class="Symbol">)</a> <a id="1489" href="Categories.Category.Core.html#656" class="Function Operator">C.∘</a> <a id="1493" href="Categories.NaturalTransformation.Core.html#783" class="Function">unit.⇒.η</a> <a id="1502" class="Symbol">(</a><a id="1503" href="Categories.Functor.Core.html#797" class="Function">R.₀</a> <a id="1507" href="Categories.Adjoint.TwoSided.html#1457" class="Bound">B</a><a id="1508" class="Symbol">)</a> <a id="1510" href="Categories.Category.Core.html#595" class="Function Operator">C.≈</a> <a id="1514" href="Categories.Category.Core.html#630" class="Function">C.id</a>
<a id="_⊣⊢_.op₁"></a><a id="1522" href="Categories.Adjoint.TwoSided.html#1522" class="Function">op₁</a> <a id="1526" class="Symbol">:</a> <a id="1528" href="Categories.Functor.Core.html#816" class="Function">R.op</a> <a id="1533" href="Categories.Adjoint.TwoSided.html#863" class="Record Operator">⊣⊢</a> <a id="1536" href="Categories.Functor.Core.html#816" class="Function">L.op</a>
<a id="1543" href="Categories.Adjoint.TwoSided.html#1522" class="Function">op₁</a> <a id="1547" class="Symbol">=</a> <a id="1549" class="Keyword">record</a>
<a id="1560" class="Symbol">{</a> <a id="1562" href="Categories.Adjoint.TwoSided.html#960" class="Field">unit</a> <a id="1569" class="Symbol">=</a> <a id="1571" href="Categories.NaturalTransformation.NaturalIsomorphism.html#1266" class="Function">counit.op</a>
<a id="1585" class="Symbol">;</a> <a id="1587" href="Categories.Adjoint.TwoSided.html#988" class="Field">counit</a> <a id="1594" class="Symbol">=</a> <a id="1596" href="Categories.NaturalTransformation.NaturalIsomorphism.html#1266" class="Function">unit.op</a>
<a id="1608" class="Symbol">;</a> <a id="1610" href="Categories.Adjoint.TwoSided.html#1373" class="Field">zig</a> <a id="1617" class="Symbol">=</a> <a id="1619" href="Categories.Adjoint.TwoSided.html#1448" class="Field">zag</a>
<a id="1627" class="Symbol">;</a> <a id="1629" href="Categories.Adjoint.TwoSided.html#1448" class="Field">zag</a> <a id="1636" class="Symbol">=</a> <a id="1638" href="Categories.Adjoint.TwoSided.html#1373" class="Field">zig</a>
<a id="1646" class="Symbol">}</a>
<a id="_⊣⊢_.zag⁻¹"></a><a id="1651" href="Categories.Adjoint.TwoSided.html#1651" class="Function">zag⁻¹</a> <a id="1657" class="Symbol">:</a> <a id="1659" class="Symbol">{</a><a id="1660" href="Categories.Adjoint.TwoSided.html#1660" class="Bound">B</a> <a id="1662" class="Symbol">:</a> <a id="1664" href="Categories.Category.Core.html#559" class="Function">D.Obj</a><a id="1669" class="Symbol">}</a> <a id="1671" class="Symbol">→</a> <a id="1673" href="Categories.NaturalTransformation.Core.html#783" class="Function">unit.⇐.η</a> <a id="1682" class="Symbol">(</a><a id="1683" href="Categories.Functor.Core.html#797" class="Function">R.₀</a> <a id="1687" href="Categories.Adjoint.TwoSided.html#1660" class="Bound">B</a><a id="1688" class="Symbol">)</a> <a id="1690" href="Categories.Category.Core.html#656" class="Function Operator">C.∘</a> <a id="1694" href="Categories.Functor.Core.html#806" class="Function">R.₁</a> <a id="1698" class="Symbol">(</a><a id="1699" href="Categories.NaturalTransformation.Core.html#783" class="Function">counit.⇐.η</a> <a id="1710" href="Categories.Adjoint.TwoSided.html#1660" class="Bound">B</a><a id="1711" class="Symbol">)</a> <a id="1713" href="Categories.Category.Core.html#595" class="Function Operator">C.≈</a> <a id="1717" href="Categories.Category.Core.html#630" class="Function">C.id</a>
<a id="1724" href="Categories.Adjoint.TwoSided.html#1651" class="Function">zag⁻¹</a> <a id="1730" class="Symbol">{</a><a id="1731" href="Categories.Adjoint.TwoSided.html#1731" class="Bound">B</a><a id="1732" class="Symbol">}</a> <a id="1734" class="Symbol">=</a> <a id="1736" href="Relation.Binary.Reasoning.Base.Single.html#1925" class="Function Operator">begin</a>
<a id="1746" href="Categories.NaturalTransformation.Core.html#783" class="Function">unit.⇐.η</a> <a id="1755" class="Symbol">(</a><a id="1756" href="Categories.Functor.Core.html#797" class="Function">R.₀</a> <a id="1760" href="Categories.Adjoint.TwoSided.html#1731" class="Bound">B</a><a id="1761" class="Symbol">)</a> <a id="1763" href="Categories.Category.Core.html#656" class="Function Operator">C.∘</a> <a id="1767" href="Categories.Functor.Core.html#806" class="Function">R.₁</a> <a id="1771" class="Symbol">(</a><a id="1772" href="Categories.NaturalTransformation.Core.html#783" class="Function">counit.⇐.η</a> <a id="1783" href="Categories.Adjoint.TwoSided.html#1731" class="Bound">B</a><a id="1784" class="Symbol">)</a> <a id="1788" href="Relation.Binary.Reasoning.Setoid.html#1162" class="Function">≈˘⟨</a> <a id="1792" href="Categories.Morphism.Reasoning.Iso.html#2506" class="Function">flip-fromʳ</a> <a id="1803" href="Categories.NaturalTransformation.NaturalIsomorphism.html#1155" class="Function">unit.FX≅GX</a> <a id="1814" href="Categories.Adjoint.TwoSided.html#1448" class="Field">zag</a> <a id="1818" href="Categories.Category.Core.html#2837" class="Function Operator">⟩∘⟨refl</a> <a id="1826" href="Relation.Binary.Reasoning.Setoid.html#1162" class="Function">⟩</a>
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<a id="3489" class="Keyword">module</a> <a id="WithZig.R"></a><a id="3496" href="Categories.Adjoint.TwoSided.html#3496" class="Module">R</a> <a id="3498" class="Symbol">=</a> <a id="3500" href="Categories.Functor.Core.html#248" class="Module">Functor</a> <a id="3508" href="Categories.Adjoint.TwoSided.html#3041" class="Bound">R</a> <a id="3510" class="Keyword">using</a> <a id="3516" class="Symbol">(</a><a id="3517" href="Categories.Functor.Core.html#797" class="Function">₀</a><a id="3518" class="Symbol">;</a> <a id="3520" href="Categories.Functor.Core.html#806" class="Function">₁</a><a id="3521" class="Symbol">;</a> <a id="3523" href="Categories.Functor.Core.html#816" class="Function">op</a><a id="3525" class="Symbol">;</a> <a id="3527" href="Categories.Functor.Core.html#511" class="Field">identity</a><a id="3535" class="Symbol">;</a> <a id="3537" href="Categories.Functor.Core.html#696" class="Field">F-resp-≈</a><a id="3545" class="Symbol">)</a>
<a id="3552" class="Keyword">field</a>
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<a id="3715" href="Categories.Adjoint.TwoSided.html#3640" class="Function">zag</a> <a id="3719" class="Symbol">{</a><a id="3720" href="Categories.Adjoint.TwoSided.html#3720" class="Bound">B</a><a id="3721" class="Symbol">}</a> <a id="3723" class="Symbol">=</a> <a id="3725" href="Categories.NaturalTransformation.NaturalIsomorphism.Properties.html#2929" class="Function">F≃id⇒id</a> <a id="3733" class="Symbol">(</a><a id="3734" href="Categories.NaturalTransformation.NaturalIsomorphism.html#4964" class="Function">≃.sym</a> <a id="3740" href="Categories.Adjoint.TwoSided.html#3118" class="Field">unit</a><a id="3744" class="Symbol">)</a> <a id="3746" href="Categories.Adjoint.TwoSided.html#3814" class="Function">helper</a>
<a id="3759" class="Keyword">where</a> <a id="3765" class="Keyword">open</a> <a id="3770" href="Categories.Adjoint.TwoSided.html#3292" class="Module">C</a>
<a id="3784" class="Keyword">open</a> <a id="3789" href="Categories.Category.Core.html#2462" class="Module">HomReasoning</a>
<a id="3814" href="Categories.Adjoint.TwoSided.html#3814" class="Function">helper</a> <a id="3821" class="Symbol">:</a> <a id="3823" href="Categories.Functor.Core.html#806" class="Function">R.₁</a> <a id="3827" class="Symbol">(</a><a id="3828" href="Categories.Functor.Core.html#806" class="Function">L.₁</a> <a id="3832" class="Symbol">(</a><a id="3833" href="Categories.Functor.Core.html#806" class="Function">R.₁</a> <a id="3837" class="Symbol">(</a><a id="3838" href="Categories.NaturalTransformation.Core.html#783" class="Function">counit.⇒.η</a> <a id="3849" href="Categories.Adjoint.TwoSided.html#3720" class="Bound">B</a><a id="3850" class="Symbol">)</a> <a id="3852" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="3854" href="Categories.NaturalTransformation.Core.html#783" class="Function">unit.⇒.η</a> <a id="3863" class="Symbol">(</a><a id="3864" href="Categories.Functor.Core.html#797" class="Function">R.₀</a> <a id="3868" href="Categories.Adjoint.TwoSided.html#3720" class="Bound">B</a><a id="3869" class="Symbol">)))</a> <a id="3873" href="Categories.Category.Core.html#595" class="Function Operator">≈</a> <a id="3875" href="Categories.Category.Core.html#630" class="Function">id</a>
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<a id="4249" href="Categories.Functor.Core.html#806" class="Function">R.₁</a> <a id="4253" href="Categories.Category.Core.html#630" class="Function">D.id</a> <a id="4311" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">≈⟨</a> <a id="4314" href="Categories.Functor.Core.html#511" class="Function">R.identity</a> <a id="4325" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">⟩</a>
<a id="4341" href="Categories.Category.Core.html#630" class="Function">id</a> <a id="4403" href="Relation.Binary.Reasoning.Base.Single.html#2564" class="Function Operator">∎</a>
<a id="4408" class="Keyword">record</a> <a id="WithZag"></a><a id="4415" href="Categories.Adjoint.TwoSided.html#4415" class="Record">WithZag</a> <a id="4423" class="Symbol">(</a><a id="4424" href="Categories.Adjoint.TwoSided.html#4424" class="Bound">L</a> <a id="4426" class="Symbol">:</a> <a id="4428" href="Categories.Functor.Core.html#248" class="Record">Functor</a> <a id="4436" href="Categories.Adjoint.TwoSided.html#815" class="Generalizable">C</a> <a id="4438" href="Categories.Adjoint.TwoSided.html#817" class="Generalizable">D</a><a id="4439" class="Symbol">)</a> <a id="4441" class="Symbol">(</a><a id="4442" href="Categories.Adjoint.TwoSided.html#4442" class="Bound">R</a> <a id="4444" class="Symbol">:</a> <a id="4446" href="Categories.Functor.Core.html#248" class="Record">Functor</a> <a id="4454" href="Categories.Adjoint.TwoSided.html#817" class="Generalizable">D</a> <a id="4456" href="Categories.Adjoint.TwoSided.html#815" class="Generalizable">C</a><a id="4457" class="Symbol">)</a> <a id="4459" class="Symbol">:</a> <a id="4461" href="Agda.Primitive.html#320" class="Primitive">Set</a> <a id="4465" class="Symbol">(</a><a id="4466" href="Level.html#602" class="Function">levelOfTerm</a> <a id="4478" href="Categories.Adjoint.TwoSided.html#4424" class="Bound">L</a> <a id="4480" href="Agda.Primitive.html#804" class="Primitive Operator">⊔</a> <a id="4482" href="Level.html#602" class="Function">levelOfTerm</a> <a id="4494" href="Categories.Adjoint.TwoSided.html#4442" class="Bound">R</a><a id="4495" class="Symbol">)</a> <a id="4497" class="Keyword">where</a>
<a id="4507" class="Keyword">field</a>
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<a id="4578" class="Keyword">private</a>
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<a id="4638" class="Keyword">module</a> <a id="WithZag.counit"></a><a id="4645" href="Categories.Adjoint.TwoSided.html#4645" class="Module">counit</a> <a id="4652" class="Symbol">=</a> <a id="4654" href="Categories.NaturalTransformation.NaturalIsomorphism.html#651" class="Module">NaturalIsomorphism</a> <a id="4673" href="Categories.Adjoint.TwoSided.html#4549" class="Field">counit</a>
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<a id="id⊣⊢id"></a><a id="6204" href="Categories.Adjoint.TwoSided.html#6204" class="Function">id⊣⊢id</a> <a id="6211" class="Symbol">:</a> <a id="6213" href="Categories.Adjoint.TwoSided.html#421" class="Function">idF</a> <a id="6217" class="Symbol">{</a><a id="6218" class="Argument">C</a> <a id="6220" class="Symbol">=</a> <a id="6222" href="Categories.Adjoint.TwoSided.html#815" class="Generalizable">C</a><a id="6223" class="Symbol">}</a> <a id="6225" href="Categories.Adjoint.TwoSided.html#863" class="Record Operator">⊣⊢</a> <a id="6228" href="Categories.Adjoint.TwoSided.html#421" class="Function">idF</a>
<a id="6232" href="Categories.Adjoint.TwoSided.html#6204" class="Function">id⊣⊢id</a> <a id="6239" class="Symbol">{</a><a id="6240" class="Argument">C</a> <a id="6242" class="Symbol">=</a> <a id="6244" href="Categories.Adjoint.TwoSided.html#6244" class="Bound">C</a><a id="6245" class="Symbol">}</a> <a id="6247" class="Symbol">=</a> <a id="6249" class="Keyword">record</a>
<a id="6258" class="Symbol">{</a> <a id="6260" href="Categories.Adjoint.TwoSided.html#960" class="Field">unit</a> <a id="6267" class="Symbol">=</a> <a id="6269" href="Categories.NaturalTransformation.NaturalIsomorphism.html#4964" class="Function">≃.sym</a> <a id="6275" href="Categories.NaturalTransformation.NaturalIsomorphism.html#6406" class="Function">≃.unitor²</a>
<a id="6287" class="Symbol">;</a> <a id="6289" href="Categories.Adjoint.TwoSided.html#988" class="Field">counit</a> <a id="6296" class="Symbol">=</a> <a id="6298" href="Categories.NaturalTransformation.NaturalIsomorphism.html#6406" class="Function">≃.unitor²</a>
<a id="6310" class="Symbol">;</a> <a id="6312" href="Categories.Adjoint.TwoSided.html#1373" class="Field">zig</a> <a id="6319" class="Symbol">=</a> <a id="6321" href="Categories.Category.Core.html#1339" class="Field">identity²</a>
<a id="6333" class="Symbol">;</a> <a id="6335" href="Categories.Adjoint.TwoSided.html#1448" class="Field">zag</a> <a id="6342" class="Symbol">=</a> <a id="6344" href="Categories.Category.Core.html#1339" class="Field">identity²</a>
<a id="6356" class="Symbol">}</a>
<a id="6360" class="Keyword">where</a> <a id="6366" class="Keyword">open</a> <a id="6371" href="Categories.Category.Core.html#442" class="Module">Category</a> <a id="6380" href="Categories.Adjoint.TwoSided.html#6244" class="Bound">C</a>
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