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bsc-leon-vatthauer/agda/bsc-thesis/Monad.Instance.K.PreElgot.html
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<pre class="Agda"><a id="449" class="Keyword">module</a> <a id="456" href="Monad.Instance.K.PreElgot.html" class="Module">Monad.Instance.K.PreElgot</a> <a id="482" class="Symbol">{</a><a id="483" href="Monad.Instance.K.PreElgot.html#483" class="Bound">o</a> <a id="485" href="Monad.Instance.K.PreElgot.html#485" class="Bound">ℓ</a> <a id="487" href="Monad.Instance.K.PreElgot.html#487" class="Bound">e</a><a id="488" class="Symbol">}</a> <a id="490" class="Symbol">(</a><a id="491" href="Monad.Instance.K.PreElgot.html#491" class="Bound">ambient</a> <a id="499" class="Symbol">:</a> <a id="501" href="Category.Ambient.html#1681" class="Record">Ambient</a> <a id="509" href="Monad.Instance.K.PreElgot.html#483" class="Bound">o</a> <a id="511" href="Monad.Instance.K.PreElgot.html#485" class="Bound">ℓ</a> <a id="513" href="Monad.Instance.K.PreElgot.html#487" class="Bound">e</a><a id="514" class="Symbol">)</a> <a id="516" class="Symbol">(</a><a id="517" href="Monad.Instance.K.PreElgot.html#517" class="Bound">MK</a> <a id="520" class="Symbol">:</a> <a id="522" href="Monad.Instance.K.html#1123" class="Record">MIK.MonadK</a> <a id="533" href="Monad.Instance.K.PreElgot.html#491" class="Bound">ambient</a><a id="540" class="Symbol">)</a> <a id="542" class="Keyword">where</a>
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</pre>
<h1 id="k-is-the-initial-pre-elgot-monad">K is the initial pre-Elgot
monad</h1>
<pre class="Agda"><a id="isPreElgot"></a><a id="1025" href="Monad.Instance.K.PreElgot.html#1025" class="Function">isPreElgot</a> <a id="1036" class="Symbol">:</a> <a id="1038" href="Monad.PreElgot.html#563" class="Record">IsPreElgot</a> <a id="1049" href="Monad.Instance.K.html#1609" class="Function">monadK</a>
<a id="1056" href="Monad.Instance.K.PreElgot.html#1025" class="Function">isPreElgot</a> <a id="1067" class="Symbol">=</a> <a id="1069" class="Keyword">record</a>
<a id="1078" class="Symbol">{</a> <a id="1080" href="Monad.PreElgot.html#815" class="Field">elgotalgebras</a> <a id="1094" class="Symbol">=</a> <a id="1096" class="Symbol">λ</a> <a id="1098" class="Symbol">{</a><a id="1099" href="Monad.Instance.K.PreElgot.html#1099" class="Bound">X</a><a id="1100" class="Symbol">}</a> <a id="1102" class="Symbol">→</a> <a id="1104" href="Algebra.Elgot.html#16032" class="Field">Elgot-Algebra.algebra</a> <a id="1126" class="Symbol">(</a><a id="1127" href="Monad.Instance.K.html#1333" class="Function">algebras</a> <a id="1136" href="Monad.Instance.K.PreElgot.html#1099" class="Bound">X</a><a id="1137" class="Symbol">)</a>
<a id="1141" class="Symbol">;</a> <a id="1143" href="Monad.PreElgot.html#998" class="Field">extend-preserves</a> <a id="1160" class="Symbol">=</a> <a id="1162" class="Symbol">λ</a> <a id="1164" href="Monad.Instance.K.PreElgot.html#1164" class="Bound">f</a> <a id="1166" href="Monad.Instance.K.PreElgot.html#1166" class="Bound">h</a> <a id="1168" class="Symbol">→</a> <a id="1170" href="Relation.Binary.Structures.html#1622" class="Function">sym</a> <a id="1174" class="Symbol">(</a><a id="1175" href="Monad.Instance.K.Commutative.html#8711" class="Function">extend-preserve</a> <a id="1191" href="Monad.Instance.K.PreElgot.html#1166" class="Bound">h</a> <a id="1193" href="Monad.Instance.K.PreElgot.html#1164" class="Bound">f</a><a id="1194" class="Symbol">)</a>
<a id="1198" class="Symbol">}</a>
<a id="1202" class="Keyword">where</a> <a id="1208" class="Keyword">open</a> <a id="1213" href="Monad.Instance.K.html#1775" class="Module">kleisliK</a> <a id="1222" class="Keyword">using</a> <a id="1228" class="Symbol">(</a><a id="1229" href="Categories.Monad.Relative.html#895" class="Function">extend</a><a id="1235" class="Symbol">)</a>
<a id="preElgot"></a><a id="1238" href="Monad.Instance.K.PreElgot.html#1238" class="Function">preElgot</a> <a id="1247" class="Symbol">:</a> <a id="1249" href="Monad.PreElgot.html#1160" class="Record">PreElgotMonad</a>
<a id="1263" href="Monad.Instance.K.PreElgot.html#1238" class="Function">preElgot</a> <a id="1272" class="Symbol">=</a> <a id="1274" class="Keyword">record</a> <a id="1281" class="Symbol">{</a> <a id="1283" href="Monad.PreElgot.html#1214" class="Field">T</a> <a id="1285" class="Symbol">=</a> <a id="1287" href="Monad.Instance.K.html#1609" class="Function">monadK</a> <a id="1294" class="Symbol">;</a> <a id="1296" href="Monad.PreElgot.html#1232" class="Field">isPreElgot</a> <a id="1307" class="Symbol">=</a> <a id="1309" href="Monad.Instance.K.PreElgot.html#1025" class="Function">isPreElgot</a> <a id="1320" class="Symbol">}</a>
<a id="isInitialPreElgot"></a><a id="1323" href="Monad.Instance.K.PreElgot.html#1323" class="Function">isInitialPreElgot</a> <a id="1341" class="Symbol">:</a> <a id="1343" href="Categories.Object.Initial.html#422" class="Record">IsInitial</a> <a id="1353" href="Category.Construction.PreElgotMonads.html#1575" class="Function">PreElgotMonads</a> <a id="1368" href="Monad.Instance.K.PreElgot.html#1238" class="Function">preElgot</a>
<a id="1377" href="Monad.Instance.K.PreElgot.html#1323" class="Function">isInitialPreElgot</a> <a id="1395" class="Symbol">=</a> <a id="1397" class="Keyword">record</a>
<a id="1406" class="Symbol">{</a> <a id="1408" href="Categories.Object.Initial.html#478" class="Field">!</a> <a id="1410" class="Symbol">=</a> <a id="1412" href="Monad.Instance.K.PreElgot.html#1456" class="Function">!′</a>
<a id="1417" class="Symbol">;</a> <a id="1419" href="Categories.Object.Initial.html#506" class="Field">!-unique</a> <a id="1428" class="Symbol">=</a> <a id="1430" href="Monad.Instance.K.PreElgot.html#10601" class="Function">!-unique′</a>
<a id="1442" class="Symbol">}</a>
<a id="1446" class="Keyword">where</a>
<a id="1456" href="Monad.Instance.K.PreElgot.html#1456" class="Function">!′</a> <a id="1459" class="Symbol">:</a> <a id="1461" class="Symbol">∀</a> <a id="1463" class="Symbol">{</a><a id="1464" href="Monad.Instance.K.PreElgot.html#1464" class="Bound">A</a> <a id="1466" class="Symbol">:</a> <a id="1468" href="Monad.PreElgot.html#1160" class="Record">PreElgotMonad</a><a id="1481" class="Symbol">}</a> <a id="1483" class="Symbol">→</a> <a id="1485" href="Category.Construction.PreElgotMonads.html#1190" class="Record">PreElgotMonad-Morphism</a> <a id="1508" href="Monad.Instance.K.PreElgot.html#1238" class="Function">preElgot</a> <a id="1517" href="Monad.Instance.K.PreElgot.html#1464" class="Bound">A</a>
<a id="1523" href="Monad.Instance.K.PreElgot.html#1456" class="Function">!′</a> <a id="1526" class="Symbol">{</a><a id="1527" href="Monad.Instance.K.PreElgot.html#1527" class="Bound">A</a><a id="1528" class="Symbol">}</a> <a id="1530" class="Symbol">=</a> <a id="1532" class="Keyword">record</a>
<a id="1545" class="Symbol">{</a> <a id="1547" href="Category.Construction.PreElgotMonads.html#1253" class="Field">α</a> <a id="1549" class="Symbol">=</a> <a id="1551" href="Categories.NaturalTransformation.Core.html#1750" class="Function">ntHelper</a> <a id="1560" class="Symbol">(</a><a id="1561" class="Keyword">record</a>
<a id="1576" class="Symbol">{</a> <a id="1578" href="Categories.NaturalTransformation.Core.html#1637" class="Field">η</a> <a id="1580" class="Symbol">=</a> <a id="1582" href="Monad.Instance.K.PreElgot.html#2394" class="Function">η&#39;</a>
<a id="1593" class="Symbol">;</a> <a id="1595" href="Categories.NaturalTransformation.Core.html#1681" class="Field">commute</a> <a id="1603" class="Symbol">=</a> <a id="1605" href="Monad.Instance.K.PreElgot.html#3507" class="Function">commute</a>
<a id="1621" class="Symbol">})</a>
<a id="1630" class="Symbol">;</a> <a id="1632" href="Category.Construction.PreElgotMonads.html#1344" class="Field">α-η</a> <a id="1636" class="Symbol">=</a> <a id="1638" href="Categories.FreeObjects.Free.html#937" class="Field">FreeObject.*-lift</a> <a id="1656" class="Symbol">(</a><a id="1657" href="Monad.Instance.K.html#1182" class="Field">freealgebras</a> <a id="1670" class="Symbol">_)</a> <a id="1673" class="Symbol">(</a><a id="1674" href="Categories.NaturalTransformation.Core.html#783" class="Function">T.η.η</a> <a id="1680" class="Symbol">_)</a>
<a id="1689" class="Symbol">;</a> <a id="1691" href="Category.Construction.PreElgotMonads.html#1400" class="Field">α-μ</a> <a id="1695" class="Symbol">=</a> <a id="1697" href="Monad.Instance.K.PreElgot.html#6624" class="Function">α-μ</a>
<a id="1707" class="Symbol">;</a> <a id="1709" href="Category.Construction.PreElgotMonads.html#1490" class="Field">preserves</a> <a id="1719" class="Symbol">=</a> <a id="1721" class="Symbol">λ</a> <a id="1723" class="Symbol">{</a><a id="1724" href="Monad.Instance.K.PreElgot.html#1724" class="Bound">X</a><a id="1725" class="Symbol">}</a> <a id="1727" class="Symbol">{</a><a id="1728" href="Monad.Instance.K.PreElgot.html#1728" class="Bound">B</a><a id="1729" class="Symbol">}</a> <a id="1731" href="Monad.Instance.K.PreElgot.html#1731" class="Bound">f</a> <a id="1733" class="Symbol">→</a> <a id="1735" href="Category.Construction.ElgotAlgebras.html#812" class="Field">Elgot-Algebra-Morphism.preserves</a> <a id="1768" class="Symbol">(((</a><a id="1771" href="Monad.Instance.K.html#1182" class="Field">freealgebras</a> <a id="1784" href="Monad.Instance.K.PreElgot.html#1728" class="Bound">B</a><a id="1785" class="Symbol">)</a> <a id="1787" href="Categories.FreeObjects.Free.html#880" class="Field Operator">FreeObject.*</a><a id="1799" class="Symbol">)</a> <a id="1801" class="Symbol">{</a><a id="1802" class="Argument">A</a> <a id="1804" class="Symbol">=</a> <a id="1806" class="Keyword">record</a> <a id="1813" class="Symbol">{</a> <a id="1815" href="Algebra.Elgot.html#16018" class="Field">A</a> <a id="1817" class="Symbol">=</a> <a id="1819" href="Categories.Functor.Core.html#432" class="Function">T.F.F₀</a> <a id="1826" href="Monad.Instance.K.PreElgot.html#1728" class="Bound">B</a> <a id="1828" class="Symbol">;</a> <a id="1830" href="Algebra.Elgot.html#16032" class="Field">algebra</a> <a id="1838" class="Symbol">=</a> <a id="1840" href="Monad.PreElgot.html#815" class="Function">PreElgotMonad.elgotalgebras</a> <a id="1868" href="Monad.Instance.K.PreElgot.html#1527" class="Bound">A</a> <a id="1870" class="Symbol">}}</a> <a id="1873" class="Symbol">(</a><a id="1874" href="Categories.NaturalTransformation.Core.html#783" class="Function">T.η.η</a> <a id="1880" href="Monad.Instance.K.PreElgot.html#1728" class="Bound">B</a><a id="1881" class="Symbol">))</a>
<a id="1890" class="Symbol">}</a>
<a id="1898" class="Keyword">where</a>
<a id="1912" class="Keyword">open</a> <a id="1917" href="Monad.PreElgot.html#1160" class="Module">PreElgotMonad</a> <a id="1931" href="Monad.Instance.K.PreElgot.html#1527" class="Bound">A</a> <a id="1933" class="Keyword">using</a> <a id="1939" class="Symbol">(</a><a id="1940" href="Monad.PreElgot.html#1214" class="Field">T</a><a id="1941" class="Symbol">)</a>
<a id="1951" class="Keyword">open</a> <a id="1956" href="Monad.Instance.K.PreElgot.html#343" class="Module">RMonad</a> <a id="1963" class="Symbol">(</a><a id="1964" href="Categories.Monad.Construction.Kleisli.html#2851" class="Function">Monad⇒Kleisli</a> <a id="1978" href="Category.Ambient.html#1757" class="Function">C</a> <a id="1980" href="Monad.PreElgot.html#1214" class="Field">T</a><a id="1981" class="Symbol">)</a> <a id="1983" class="Keyword">using</a> <a id="1989" class="Symbol">(</a><a id="1990" href="Categories.Monad.Relative.html#895" class="Field">extend</a><a id="1996" class="Symbol">)</a>
<a id="2006" class="Keyword">module</a> <a id="2013" href="Monad.Instance.K.PreElgot.html#2013" class="Module">T</a> <a id="2015" class="Symbol">=</a> <a id="2017" href="Categories.Monad.html#454" class="Module">Monad</a> <a id="2023" href="Monad.PreElgot.html#1214" class="Field">T</a>
<a id="2033" class="Keyword">open</a> <a id="2038" href="Monad.Instance.K.html#1672" class="Module">monadK</a> <a id="2045" class="Keyword">using</a> <a id="2051" class="Symbol">()</a> <a id="2054" class="Keyword">renaming</a> <a id="2063" class="Symbol">(</a><a id="2064" href="Categories.Monad.html#547" class="Function">η</a> <a id="2066" class="Symbol">to</a> <a id="2069" class="Function">ηK</a><a id="2071" class="Symbol">;</a> <a id="2073" href="Categories.Monad.html#583" class="Function">μ</a> <a id="2075" class="Symbol">to</a> <a id="2078" class="Function">μK</a><a id="2080" class="Symbol">)</a>
<a id="2090" class="Keyword">open</a> <a id="2095" href="Algebra.Elgot.html#1845" class="Module">Elgot-Algebra-on</a> <a id="2112" class="Keyword">using</a> <a id="2118" class="Symbol">(</a><a id="2119" href="Algebra.Elgot.html#2331" class="Field">#-resp-≈</a><a id="2127" class="Symbol">)</a>
<a id="2137" href="Monad.Instance.K.PreElgot.html#2137" class="Function">T-Alg</a> <a id="2143" class="Symbol">:</a> <a id="2145" class="Symbol">∀</a> <a id="2147" class="Symbol">(</a><a id="2148" href="Monad.Instance.K.PreElgot.html#2148" class="Bound">X</a> <a id="2150" class="Symbol">:</a> <a id="2152" href="Categories.Category.Core.html#559" class="Function">Obj</a><a id="2155" class="Symbol">)</a> <a id="2157" class="Symbol">→</a> <a id="2159" href="Algebra.Elgot.html#15964" class="Record">Elgot-Algebra</a>
<a id="2181" href="Monad.Instance.K.PreElgot.html#2137" class="Function">T-Alg</a> <a id="2187" href="Monad.Instance.K.PreElgot.html#2187" class="Bound">X</a> <a id="2189" class="Symbol">=</a> <a id="2191" class="Keyword">record</a> <a id="2198" class="Symbol">{</a> <a id="2200" href="Algebra.Elgot.html#16018" class="Field">A</a> <a id="2202" class="Symbol">=</a> <a id="2204" href="Categories.Functor.Core.html#797" class="Function">T.F.₀</a> <a id="2210" href="Monad.Instance.K.PreElgot.html#2187" class="Bound">X</a> <a id="2212" class="Symbol">;</a> <a id="2214" href="Algebra.Elgot.html#16032" class="Field">algebra</a> <a id="2222" class="Symbol">=</a> <a id="2224" href="Monad.PreElgot.html#815" class="Function">PreElgotMonad.elgotalgebras</a> <a id="2252" href="Monad.Instance.K.PreElgot.html#1527" class="Bound">A</a> <a id="2254" class="Symbol">}</a>
<a id="2264" href="Monad.Instance.K.PreElgot.html#2264" class="Function">K-Alg</a> <a id="2270" class="Symbol">:</a> <a id="2272" class="Symbol">∀</a> <a id="2274" class="Symbol">(</a><a id="2275" href="Monad.Instance.K.PreElgot.html#2275" class="Bound">X</a> <a id="2277" class="Symbol">:</a> <a id="2279" href="Categories.Category.Core.html#559" class="Function">Obj</a><a id="2282" class="Symbol">)</a> <a id="2284" class="Symbol">→</a> <a id="2286" href="Algebra.Elgot.html#15964" class="Record">Elgot-Algebra</a>
<a id="2308" href="Monad.Instance.K.PreElgot.html#2264" class="Function">K-Alg</a> <a id="2314" href="Monad.Instance.K.PreElgot.html#2314" class="Bound">X</a> <a id="2316" class="Symbol">=</a> <a id="2318" class="Keyword">record</a> <a id="2325" class="Symbol">{</a> <a id="2327" href="Algebra.Elgot.html#16018" class="Field">A</a> <a id="2329" class="Symbol">=</a> <a id="2331" href="Categories.Functor.Core.html#797" class="Function">K.₀</a> <a id="2335" href="Monad.Instance.K.PreElgot.html#2314" class="Bound">X</a> <a id="2337" class="Symbol">;</a> <a id="2339" href="Algebra.Elgot.html#16032" class="Field">algebra</a> <a id="2347" class="Symbol">=</a> <a id="2349" href="Algebra.Elgot.html#16032" class="Field">Elgot-Algebra.algebra</a> <a id="2371" class="Symbol">(</a><a id="2372" href="Monad.Instance.K.html#1333" class="Function">algebras</a> <a id="2381" href="Monad.Instance.K.PreElgot.html#2314" class="Bound">X</a><a id="2382" class="Symbol">)</a> <a id="2384" class="Symbol">}</a>
<a id="2394" href="Monad.Instance.K.PreElgot.html#2394" class="Function">η&#39;</a> <a id="2397" class="Symbol">:</a> <a id="2399" class="Symbol">∀</a> <a id="2401" class="Symbol">(</a><a id="2402" href="Monad.Instance.K.PreElgot.html#2402" class="Bound">X</a> <a id="2404" class="Symbol">:</a> <a id="2406" href="Categories.Category.Core.html#559" class="Function">Obj</a><a id="2409" class="Symbol">)</a> <a id="2411" class="Symbol">→</a> <a id="2413" href="Categories.Functor.Core.html#797" class="Function">K.₀</a> <a id="2417" href="Monad.Instance.K.PreElgot.html#2402" class="Bound">X</a> <a id="2419" href="Categories.Category.Core.html#575" class="Function Operator">⇒</a> <a id="2421" href="Categories.Functor.Core.html#797" class="Function">T.F.₀</a> <a id="2427" href="Monad.Instance.K.PreElgot.html#2402" class="Bound">X</a>
<a id="2437" href="Monad.Instance.K.PreElgot.html#2394" class="Function">η&#39;</a> <a id="2440" href="Monad.Instance.K.PreElgot.html#2440" class="Bound">X</a> <a id="2442" class="Symbol">=</a> <a id="2444" href="Category.Construction.ElgotAlgebras.html#794" class="Field">Elgot-Algebra-Morphism.h</a> <a id="2469" class="Symbol">(</a><a id="2470" href="Categories.FreeObjects.Free.html#880" class="Function Operator">_*</a> <a id="2473" class="Symbol">{</a><a id="2474" class="Argument">A</a> <a id="2476" class="Symbol">=</a> <a id="2478" href="Monad.Instance.K.PreElgot.html#2137" class="Function">T-Alg</a> <a id="2484" href="Monad.Instance.K.PreElgot.html#2440" class="Bound">X</a><a id="2485" class="Symbol">}</a> <a id="2487" class="Symbol">(</a><a id="2488" href="Categories.NaturalTransformation.Core.html#783" class="Function">T.η.η</a> <a id="2494" href="Monad.Instance.K.PreElgot.html#2440" class="Bound">X</a><a id="2495" class="Symbol">))</a>
<a id="2508" class="Keyword">where</a> <a id="2514" class="Keyword">open</a> <a id="2519" href="Categories.FreeObjects.Free.html#485" class="Module">FreeObject</a> <a id="2530" class="Symbol">(</a><a id="2531" href="Monad.Instance.K.html#1182" class="Field">freealgebras</a> <a id="2544" href="Monad.Instance.K.PreElgot.html#2440" class="Bound">X</a><a id="2545" class="Symbol">)</a>
<a id="2555" href="Monad.Instance.K.PreElgot.html#2555" class="Function Operator">_#K</a> <a id="2559" class="Symbol">=</a> <a id="2561" class="Symbol">λ</a> <a id="2563" class="Symbol">{</a><a id="2564" href="Monad.Instance.K.PreElgot.html#2564" class="Bound">B</a><a id="2565" class="Symbol">}</a> <a id="2567" class="Symbol">{</a><a id="2568" href="Monad.Instance.K.PreElgot.html#2568" class="Bound">C</a><a id="2569" class="Symbol">}</a> <a id="2571" href="Monad.Instance.K.PreElgot.html#2571" class="Bound">f</a> <a id="2573" class="Symbol">→</a> <a id="2575" href="Algebra.Elgot.html#1938" class="Function Operator">Elgot-Algebra._#</a> <a id="2592" class="Symbol">(</a><a id="2593" href="Categories.FreeObjects.Free.html#838" class="Field">FreeObject.FX</a> <a id="2607" class="Symbol">(</a><a id="2608" href="Monad.Instance.K.html#1182" class="Field">freealgebras</a> <a id="2621" href="Monad.Instance.K.PreElgot.html#2568" class="Bound">C</a><a id="2622" class="Symbol">))</a> <a id="2625" class="Symbol">{</a><a id="2626" href="Monad.Instance.K.PreElgot.html#2564" class="Bound">B</a><a id="2627" class="Symbol">}</a> <a id="2629" href="Monad.Instance.K.PreElgot.html#2571" class="Bound">f</a>
<a id="2639" href="Monad.Instance.K.PreElgot.html#2639" class="Function Operator">_#T</a> <a id="2643" class="Symbol">=</a> <a id="2645" class="Symbol">λ</a> <a id="2647" class="Symbol">{</a><a id="2648" href="Monad.Instance.K.PreElgot.html#2648" class="Bound">B</a><a id="2649" class="Symbol">}</a> <a id="2651" class="Symbol">{</a><a id="2652" href="Monad.Instance.K.PreElgot.html#2652" class="Bound">C</a><a id="2653" class="Symbol">}</a> <a id="2655" href="Monad.Instance.K.PreElgot.html#2655" class="Bound">f</a> <a id="2657" class="Symbol">→</a> <a id="2659" href="Algebra.Elgot.html#1938" class="Function Operator">PreElgotMonad.elgotalgebras._#</a> <a id="2690" href="Monad.Instance.K.PreElgot.html#1527" class="Bound">A</a> <a id="2692" class="Symbol">{</a><a id="2693" href="Monad.Instance.K.PreElgot.html#2648" class="Bound">B</a><a id="2694" class="Symbol">}</a> <a id="2696" class="Symbol">{</a><a id="2697" href="Monad.Instance.K.PreElgot.html#2652" class="Bound">C</a><a id="2698" class="Symbol">}</a> <a id="2700" href="Monad.Instance.K.PreElgot.html#2655" class="Bound">f</a>
<a id="2710" class="Comment">-- some preservation facts that follow immediately, since these things are elgot-algebra-morphisms.</a>
<a id="2818" href="Monad.Instance.K.PreElgot.html#2818" class="Function">K₁-preserves</a> <a id="2831" class="Symbol">:</a> <a id="2833" class="Symbol">∀</a> <a id="2835" class="Symbol">{</a><a id="2836" href="Monad.Instance.K.PreElgot.html#2836" class="Bound">X</a> <a id="2838" href="Monad.Instance.K.PreElgot.html#2838" class="Bound">Y</a> <a id="2840" href="Monad.Instance.K.PreElgot.html#2840" class="Bound">Z</a> <a id="2842" class="Symbol">:</a> <a id="2844" href="Categories.Category.Core.html#559" class="Function">Obj</a><a id="2847" class="Symbol">}</a> <a id="2849" class="Symbol">(</a><a id="2850" href="Monad.Instance.K.PreElgot.html#2850" class="Bound">f</a> <a id="2852" class="Symbol">:</a> <a id="2854" href="Monad.Instance.K.PreElgot.html#2836" class="Bound">X</a> <a id="2856" href="Categories.Category.Core.html#575" class="Function Operator">⇒</a> <a id="2858" href="Monad.Instance.K.PreElgot.html#2838" class="Bound">Y</a><a id="2859" class="Symbol">)</a> <a id="2861" class="Symbol">(</a><a id="2862" href="Monad.Instance.K.PreElgot.html#2862" class="Bound">g</a> <a id="2864" class="Symbol">:</a> <a id="2866" href="Monad.Instance.K.PreElgot.html#2840" class="Bound">Z</a> <a id="2868" href="Categories.Category.Core.html#575" class="Function Operator">⇒</a> <a id="2870" href="Categories.Functor.Core.html#797" class="Function">K.₀</a> <a id="2874" href="Monad.Instance.K.PreElgot.html#2836" class="Bound">X</a> <a id="2876" href="Categories.Category.Cocartesian.html#1549" class="Function Operator">+</a> <a id="2878" href="Monad.Instance.K.PreElgot.html#2840" class="Bound">Z</a><a id="2879" class="Symbol">)</a> <a id="2881" class="Symbol">→</a> <a id="2883" href="Categories.Functor.Core.html#806" class="Function">K.₁</a> <a id="2887" href="Monad.Instance.K.PreElgot.html#2850" class="Bound">f</a> <a id="2889" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="2891" class="Symbol">(</a><a id="2892" href="Monad.Instance.K.PreElgot.html#2862" class="Bound">g</a> <a id="2894" href="Monad.Instance.K.PreElgot.html#2555" class="Function Operator">#K</a><a id="2896" class="Symbol">)</a> <a id="2898" href="Categories.Category.Core.html#595" class="Function Operator">≈</a> <a id="2900" class="Symbol">((</a><a id="2902" href="Categories.Functor.Core.html#806" class="Function">K.₁</a> <a id="2906" href="Monad.Instance.K.PreElgot.html#2850" class="Bound">f</a> <a id="2908" href="Categories.Category.Cocartesian.html#2133" class="Function Operator">+₁</a> <a id="2911" href="Category.Ambient.html#2115" class="Function">idC</a><a id="2914" class="Symbol">)</a> <a id="2916" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="2918" href="Monad.Instance.K.PreElgot.html#2862" class="Bound">g</a><a id="2919" class="Symbol">)</a> <a id="2921" href="Monad.Instance.K.PreElgot.html#2555" class="Function Operator">#K</a>
<a id="2932" href="Monad.Instance.K.PreElgot.html#2818" class="Function">K₁-preserves</a> <a id="2945" class="Symbol">{</a><a id="2946" href="Monad.Instance.K.PreElgot.html#2946" class="Bound">X</a><a id="2947" class="Symbol">}</a> <a id="2949" class="Symbol">{</a><a id="2950" href="Monad.Instance.K.PreElgot.html#2950" class="Bound">Y</a><a id="2951" class="Symbol">}</a> <a id="2953" class="Symbol">{</a><a id="2954" href="Monad.Instance.K.PreElgot.html#2954" class="Bound">Z</a><a id="2955" class="Symbol">}</a> <a id="2957" href="Monad.Instance.K.PreElgot.html#2957" class="Bound">f</a> <a id="2959" href="Monad.Instance.K.PreElgot.html#2959" class="Bound">g</a> <a id="2961" class="Symbol">=</a> <a id="2963" href="Category.Construction.ElgotAlgebras.html#812" class="Field">Elgot-Algebra-Morphism.preserves</a> <a id="2996" class="Symbol">(((</a><a id="2999" href="Monad.Instance.K.html#1182" class="Field">freealgebras</a> <a id="3012" href="Monad.Instance.K.PreElgot.html#2946" class="Bound">X</a><a id="3013" class="Symbol">)</a> <a id="3015" href="Categories.FreeObjects.Free.html#880" class="Field Operator">FreeObject.*</a><a id="3027" class="Symbol">)</a> <a id="3029" class="Symbol">{</a><a id="3030" class="Argument">A</a> <a id="3032" class="Symbol">=</a> <a id="3034" href="Monad.Instance.K.PreElgot.html#2264" class="Function">K-Alg</a> <a id="3040" href="Monad.Instance.K.PreElgot.html#2950" class="Bound">Y</a><a id="3041" class="Symbol">}</a> <a id="3043" class="Symbol">(</a><a id="3044" href="Categories.NaturalTransformation.Core.html#783" class="Function">ηK.η</a> <a id="3049" class="Symbol">_</a> <a id="3051" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="3053" href="Monad.Instance.K.PreElgot.html#2957" class="Bound">f</a><a id="3054" class="Symbol">))</a>
<a id="3065" href="Monad.Instance.K.PreElgot.html#3065" class="Function">μK-preserves</a> <a id="3078" class="Symbol">:</a> <a id="3080" class="Symbol">∀</a> <a id="3082" class="Symbol">{</a><a id="3083" href="Monad.Instance.K.PreElgot.html#3083" class="Bound">X</a> <a id="3085" href="Monad.Instance.K.PreElgot.html#3085" class="Bound">Y</a> <a id="3087" class="Symbol">:</a> <a id="3089" href="Categories.Category.Core.html#559" class="Function">Obj</a><a id="3092" class="Symbol">}</a> <a id="3094" class="Symbol">(</a><a id="3095" href="Monad.Instance.K.PreElgot.html#3095" class="Bound">g</a> <a id="3097" class="Symbol">:</a> <a id="3099" href="Monad.Instance.K.PreElgot.html#3085" class="Bound">Y</a> <a id="3101" href="Categories.Category.Core.html#575" class="Function Operator">⇒</a> <a id="3103" href="Categories.Functor.Core.html#797" class="Function">K.₀</a> <a id="3107" class="Symbol">(</a><a id="3108" href="Categories.Functor.Core.html#797" class="Function">K.₀</a> <a id="3112" href="Monad.Instance.K.PreElgot.html#3083" class="Bound">X</a><a id="3113" class="Symbol">)</a> <a id="3115" href="Categories.Category.Cocartesian.html#1549" class="Function Operator">+</a> <a id="3117" href="Monad.Instance.K.PreElgot.html#3085" class="Bound">Y</a><a id="3118" class="Symbol">)</a> <a id="3120" class="Symbol">→</a> <a id="3122" href="Categories.NaturalTransformation.Core.html#783" class="Function">μK.η</a> <a id="3127" href="Monad.Instance.K.PreElgot.html#3083" class="Bound">X</a> <a id="3129" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="3131" href="Monad.Instance.K.PreElgot.html#3095" class="Bound">g</a> <a id="3133" href="Monad.Instance.K.PreElgot.html#2555" class="Function Operator">#K</a> <a id="3136" href="Categories.Category.Core.html#595" class="Function Operator">≈</a> <a id="3138" class="Symbol">((</a><a id="3140" href="Categories.NaturalTransformation.Core.html#783" class="Function">μK.η</a> <a id="3145" href="Monad.Instance.K.PreElgot.html#3083" class="Bound">X</a> <a id="3147" href="Categories.Category.Cocartesian.html#2133" class="Function Operator">+₁</a> <a id="3150" href="Category.Ambient.html#2115" class="Function">idC</a><a id="3153" class="Symbol">)</a> <a id="3155" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="3157" href="Monad.Instance.K.PreElgot.html#3095" class="Bound">g</a><a id="3158" class="Symbol">)</a> <a id="3160" href="Monad.Instance.K.PreElgot.html#2555" class="Function Operator">#K</a>
<a id="3171" href="Monad.Instance.K.PreElgot.html#3065" class="Function">μK-preserves</a> <a id="3184" class="Symbol">{</a><a id="3185" href="Monad.Instance.K.PreElgot.html#3185" class="Bound">X</a><a id="3186" class="Symbol">}</a> <a id="3188" href="Monad.Instance.K.PreElgot.html#3188" class="Bound">g</a> <a id="3190" class="Symbol">=</a> <a id="3192" href="Category.Construction.ElgotAlgebras.html#812" class="Field">Elgot-Algebra-Morphism.preserves</a> <a id="3225" class="Symbol">(((</a><a id="3228" href="Monad.Instance.K.html#1182" class="Field">freealgebras</a> <a id="3241" class="Symbol">(</a><a id="3242" href="Categories.Functor.Core.html#797" class="Function">K.₀</a> <a id="3246" href="Monad.Instance.K.PreElgot.html#3185" class="Bound">X</a><a id="3247" class="Symbol">))</a> <a id="3250" href="Categories.FreeObjects.Free.html#880" class="Field Operator">FreeObject.*</a><a id="3262" class="Symbol">)</a> <a id="3264" class="Symbol">{</a><a id="3265" class="Argument">A</a> <a id="3267" class="Symbol">=</a> <a id="3269" href="Monad.Instance.K.PreElgot.html#2264" class="Function">K-Alg</a> <a id="3275" href="Monad.Instance.K.PreElgot.html#3185" class="Bound">X</a><a id="3276" class="Symbol">}</a> <a id="3278" href="Category.Ambient.html#2115" class="Function">idC</a><a id="3281" class="Symbol">)</a>
<a id="3291" href="Monad.Instance.K.PreElgot.html#3291" class="Function">η&#39;-preserves</a> <a id="3304" class="Symbol">:</a> <a id="3306" class="Symbol">∀</a> <a id="3308" class="Symbol">{</a><a id="3309" href="Monad.Instance.K.PreElgot.html#3309" class="Bound">X</a> <a id="3311" href="Monad.Instance.K.PreElgot.html#3311" class="Bound">Y</a> <a id="3313" class="Symbol">:</a> <a id="3315" href="Categories.Category.Core.html#559" class="Function">Obj</a><a id="3318" class="Symbol">}</a> <a id="3320" class="Symbol">(</a><a id="3321" href="Monad.Instance.K.PreElgot.html#3321" class="Bound">g</a> <a id="3323" class="Symbol">:</a> <a id="3325" href="Monad.Instance.K.PreElgot.html#3311" class="Bound">Y</a> <a id="3327" href="Categories.Category.Core.html#575" class="Function Operator">⇒</a> <a id="3329" href="Categories.Functor.Core.html#797" class="Function">K.₀</a> <a id="3333" href="Monad.Instance.K.PreElgot.html#3309" class="Bound">X</a> <a id="3335" href="Categories.Category.Cocartesian.html#1549" class="Function Operator">+</a> <a id="3337" href="Monad.Instance.K.PreElgot.html#3311" class="Bound">Y</a><a id="3338" class="Symbol">)</a> <a id="3340" class="Symbol">→</a> <a id="3342" href="Monad.Instance.K.PreElgot.html#2394" class="Function">η&#39;</a> <a id="3345" href="Monad.Instance.K.PreElgot.html#3309" class="Bound">X</a> <a id="3347" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="3349" href="Monad.Instance.K.PreElgot.html#3321" class="Bound">g</a> <a id="3351" href="Monad.Instance.K.PreElgot.html#2555" class="Function Operator">#K</a> <a id="3354" href="Categories.Category.Core.html#595" class="Function Operator">≈</a> <a id="3356" class="Symbol">((</a><a id="3358" href="Monad.Instance.K.PreElgot.html#2394" class="Function">η&#39;</a> <a id="3361" href="Monad.Instance.K.PreElgot.html#3309" class="Bound">X</a> <a id="3363" href="Categories.Category.Cocartesian.html#2133" class="Function Operator">+₁</a> <a id="3366" href="Category.Ambient.html#2115" class="Function">idC</a><a id="3369" class="Symbol">)</a> <a id="3371" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="3373" href="Monad.Instance.K.PreElgot.html#3321" class="Bound">g</a><a id="3374" class="Symbol">)</a> <a id="3376" href="Monad.Instance.K.PreElgot.html#2639" class="Function Operator">#T</a>
<a id="3387" href="Monad.Instance.K.PreElgot.html#3291" class="Function">η&#39;-preserves</a> <a id="3400" class="Symbol">{</a><a id="3401" href="Monad.Instance.K.PreElgot.html#3401" class="Bound">X</a><a id="3402" class="Symbol">}</a> <a id="3404" href="Monad.Instance.K.PreElgot.html#3404" class="Bound">g</a> <a id="3406" class="Symbol">=</a> <a id="3408" href="Category.Construction.ElgotAlgebras.html#812" class="Field">Elgot-Algebra-Morphism.preserves</a> <a id="3441" class="Symbol">(((</a><a id="3444" href="Monad.Instance.K.html#1182" class="Field">freealgebras</a> <a id="3457" href="Monad.Instance.K.PreElgot.html#3401" class="Bound">X</a><a id="3458" class="Symbol">)</a> <a id="3460" href="Categories.FreeObjects.Free.html#880" class="Field Operator">FreeObject.*</a><a id="3472" class="Symbol">)</a> <a id="3474" class="Symbol">{</a><a id="3475" class="Argument">A</a> <a id="3477" class="Symbol">=</a> <a id="3479" href="Monad.Instance.K.PreElgot.html#2137" class="Function">T-Alg</a> <a id="3485" href="Monad.Instance.K.PreElgot.html#3401" class="Bound">X</a><a id="3486" class="Symbol">}</a> <a id="3488" class="Symbol">(</a><a id="3489" href="Categories.NaturalTransformation.Core.html#783" class="Function">T.η.η</a> <a id="3495" href="Monad.Instance.K.PreElgot.html#3401" class="Bound">X</a><a id="3496" class="Symbol">))</a>
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<a id="5650" href="Monad.Instance.K.PreElgot.html#5543" class="Function">pres₂</a> <a id="5656" class="Symbol">{</a><a id="5657" href="Monad.Instance.K.PreElgot.html#5657" class="Bound">Z</a><a id="5658" class="Symbol">}</a> <a id="5660" class="Symbol">{</a><a id="5661" href="Monad.Instance.K.PreElgot.html#5661" class="Bound">g</a><a id="5662" class="Symbol">}</a> <a id="5664" class="Symbol">=</a> <a id="5666" href="Relation.Binary.Reasoning.Syntax.html#1510" class="Function Operator">begin</a>
<a id="5686" class="Symbol">(</a><a id="5687" href="Categories.Functor.Core.html#806" class="Function">T.F.₁</a> <a id="5693" href="Monad.Instance.K.PreElgot.html#3599" class="Bound">f</a> <a id="5695" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="5697" href="Monad.Instance.K.PreElgot.html#2394" class="Function">η&#39;</a> <a id="5700" href="Monad.Instance.K.PreElgot.html#3592" class="Bound">X</a><a id="5701" class="Symbol">)</a> <a id="5703" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="5705" href="Monad.Instance.K.PreElgot.html#5661" class="Bound">g</a> <a id="5707" href="Monad.Instance.K.PreElgot.html#2555" class="Function Operator">#K</a> <a id="5743" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="5746" href="Categories.Morphism.Reasoning.Core.html#1914" class="Function">pullʳ</a> <a id="5752" class="Symbol">(</a><a id="5753" href="Monad.Instance.K.PreElgot.html#3291" class="Function">η&#39;-preserves</a> <a id="5766" href="Monad.Instance.K.PreElgot.html#5661" class="Bound">g</a><a id="5767" class="Symbol">)</a> <a id="5769" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
<a id="5785" href="Categories.Functor.Core.html#806" class="Function">T.F.₁</a> <a id="5791" href="Monad.Instance.K.PreElgot.html#3599" class="Bound">f</a> <a id="5793" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="5795" class="Symbol">((</a><a id="5797" href="Monad.Instance.K.PreElgot.html#2394" class="Function">η&#39;</a> <a id="5800" href="Monad.Instance.K.PreElgot.html#3592" class="Bound">X</a> <a id="5802" href="Categories.Category.Cocartesian.html#2133" class="Function Operator">+₁</a> <a id="5805" href="Category.Ambient.html#2115" class="Function">idC</a><a id="5808" class="Symbol">)</a> <a id="5810" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="5812" href="Monad.Instance.K.PreElgot.html#5661" class="Bound">g</a><a id="5813" class="Symbol">)</a> <a id="5815" href="Monad.Instance.K.PreElgot.html#2639" class="Function Operator">#T</a> <a id="5842" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="5845" class="Symbol">(</a><a id="5846" href="Relation.Binary.Structures.html#1622" class="Function">sym</a> <a id="5850" class="Symbol">(</a><a id="5851" href="Category.Ambient.html#10091" class="Function">F₁⇒extend</a> <a id="5861" href="Monad.PreElgot.html#1214" class="Field">T</a> <a id="5863" href="Monad.Instance.K.PreElgot.html#3599" class="Bound">f</a><a id="5864" class="Symbol">))</a> <a id="5867" href="Categories.Category.Core.html#2837" class="Function Operator">⟩∘⟨refl</a> <a id="5875" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
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<a id="9953" href="Monad.Instance.K.PreElgot.html#9953" class="Function">comm</a> <a id="9958" class="Symbol">:</a> <a id="9960" class="Symbol">(</a><a id="9961" href="Categories.NaturalTransformation.Core.html#783" class="Function">T.μ.η</a> <a id="9967" href="Monad.Instance.K.PreElgot.html#6709" class="Bound">X</a> <a id="9969" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="9971" href="Categories.Functor.Core.html#806" class="Function">T.F.₁</a> <a id="9977" class="Symbol">(</a><a id="9978" href="Monad.Instance.K.PreElgot.html#2394" class="Function">η&#39;</a> <a id="9981" href="Monad.Instance.K.PreElgot.html#6709" class="Bound">X</a><a id="9982" class="Symbol">)</a> <a id="9984" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="9986" href="Monad.Instance.K.PreElgot.html#2394" class="Function">η&#39;</a> <a id="9989" class="Symbol">(</a><a id="9990" href="Categories.Functor.Core.html#797" class="Function">K.₀</a> <a id="9994" href="Monad.Instance.K.PreElgot.html#6709" class="Bound">X</a><a id="9995" class="Symbol">))</a> <a id="9998" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="10000" href="Categories.NaturalTransformation.Core.html#783" class="Function">ηK.η</a> <a id="10005" class="Symbol">(</a><a id="10006" href="Categories.Functor.Core.html#797" class="Function">K.₀</a> <a id="10010" href="Monad.Instance.K.PreElgot.html#6709" class="Bound">X</a><a id="10011" class="Symbol">)</a> <a id="10013" href="Categories.Category.Core.html#595" class="Function Operator">≈</a> <a id="10015" href="Monad.Instance.K.PreElgot.html#2394" class="Function">η&#39;</a> <a id="10018" href="Monad.Instance.K.PreElgot.html#6709" class="Bound">X</a>
<a id="10032" href="Monad.Instance.K.PreElgot.html#9953" class="Function">comm</a> <a id="10037" class="Symbol">=</a> <a id="10039" href="Relation.Binary.Reasoning.Syntax.html#1510" class="Function Operator">begin</a>
<a id="10059" class="Symbol">(</a><a id="10060" href="Categories.NaturalTransformation.Core.html#783" class="Function">T.μ.η</a> <a id="10066" href="Monad.Instance.K.PreElgot.html#6709" class="Bound">X</a> <a id="10068" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="10070" href="Categories.Functor.Core.html#806" class="Function">T.F.₁</a> <a id="10076" class="Symbol">(</a><a id="10077" href="Monad.Instance.K.PreElgot.html#2394" class="Function">η&#39;</a> <a id="10080" href="Monad.Instance.K.PreElgot.html#6709" class="Bound">X</a><a id="10081" class="Symbol">)</a> <a id="10083" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="10085" href="Monad.Instance.K.PreElgot.html#2394" class="Function">η&#39;</a> <a id="10088" class="Symbol">(</a><a id="10089" href="Categories.Functor.Core.html#797" class="Function">K.₀</a> <a id="10093" href="Monad.Instance.K.PreElgot.html#6709" class="Bound">X</a><a id="10094" class="Symbol">))</a> <a id="10097" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="10099" href="Categories.NaturalTransformation.Core.html#783" class="Function">ηK.η</a> <a id="10104" class="Symbol">(</a><a id="10105" href="Categories.Functor.Core.html#797" class="Function">K.₀</a> <a id="10109" href="Monad.Instance.K.PreElgot.html#6709" class="Bound">X</a><a id="10110" class="Symbol">)</a> <a id="10112" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="10115" class="Symbol">(</a><a id="10116" href="Categories.Category.Core.html#2734" class="Function Operator">refl⟩∘⟨</a> <a id="10124" href="Relation.Binary.Structures.html#1622" class="Function">sym</a> <a id="10128" class="Symbol">(</a><a id="10129" href="Monad.Instance.K.PreElgot.html#3507" class="Function">commute</a> <a id="10137" class="Symbol">(</a><a id="10138" href="Monad.Instance.K.PreElgot.html#2394" class="Function">η&#39;</a> <a id="10141" href="Monad.Instance.K.PreElgot.html#6709" class="Bound">X</a><a id="10142" class="Symbol">)))</a> <a id="10146" href="Categories.Category.Core.html#2837" class="Function Operator">⟩∘⟨refl</a> <a id="10154" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
<a id="10170" class="Symbol">(</a><a id="10171" href="Categories.NaturalTransformation.Core.html#783" class="Function">T.μ.η</a> <a id="10177" href="Monad.Instance.K.PreElgot.html#6709" class="Bound">X</a> <a id="10179" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="10181" href="Monad.Instance.K.PreElgot.html#2394" class="Function">η&#39;</a> <a id="10184" class="Symbol">_</a> <a id="10186" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="10188" href="Categories.Functor.Core.html#806" class="Function">K.₁</a> <a id="10192" class="Symbol">(</a><a id="10193" href="Monad.Instance.K.PreElgot.html#2394" class="Function">η&#39;</a> <a id="10196" href="Monad.Instance.K.PreElgot.html#6709" class="Bound">X</a><a id="10197" class="Symbol">))</a> <a id="10200" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="10202" href="Categories.NaturalTransformation.Core.html#783" class="Function">ηK.η</a> <a id="10207" class="Symbol">(</a><a id="10208" href="Categories.Functor.Core.html#797" class="Function">K.₀</a> <a id="10212" href="Monad.Instance.K.PreElgot.html#6709" class="Bound">X</a><a id="10213" class="Symbol">)</a> <a id="10223" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="10226" href="Categories.Category.Core.html#715" class="Function">assoc</a> <a id="10232" href="Categories.Category.Core.html#3061" class="Function Operator">○</a> <a id="10234" href="Categories.Category.Core.html#2734" class="Function Operator">refl⟩∘⟨</a> <a id="10242" class="Symbol">(</a><a id="10243" href="Categories.Category.Core.html#715" class="Function">assoc</a> <a id="10249" href="Categories.Category.Core.html#3061" class="Function Operator">○</a> <a id="10252" href="Categories.Category.Core.html#2734" class="Function Operator">refl⟩∘⟨</a> <a id="10260" href="Relation.Binary.Structures.html#1622" class="Function">sym</a> <a id="10264" class="Symbol">(</a><a id="10265" href="Categories.NaturalTransformation.Core.html#827" class="Function">monadK.η.commute</a> <a id="10282" class="Symbol">(</a><a id="10283" href="Monad.Instance.K.PreElgot.html#2394" class="Function">η&#39;</a> <a id="10286" href="Monad.Instance.K.PreElgot.html#6709" class="Bound">X</a><a id="10287" class="Symbol">)))</a> <a id="10291" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
<a id="10307" href="Categories.NaturalTransformation.Core.html#783" class="Function">T.μ.η</a> <a id="10313" href="Monad.Instance.K.PreElgot.html#6709" class="Bound">X</a> <a id="10315" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="10317" href="Monad.Instance.K.PreElgot.html#2394" class="Function">η&#39;</a> <a id="10320" class="Symbol">_</a> <a id="10322" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="10324" href="Categories.NaturalTransformation.Core.html#783" class="Function">ηK.η</a> <a id="10329" class="Symbol">(</a><a id="10330" href="Categories.Functor.Core.html#432" class="Function">T.F.F₀</a> <a id="10337" href="Monad.Instance.K.PreElgot.html#6709" class="Bound">X</a><a id="10338" class="Symbol">)</a> <a id="10340" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="10342" href="Monad.Instance.K.PreElgot.html#2394" class="Function">η&#39;</a> <a id="10345" href="Monad.Instance.K.PreElgot.html#6709" class="Bound">X</a> <a id="10360" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="10363" href="Categories.Category.Core.html#2734" class="Function Operator">refl⟩∘⟨</a> <a id="10371" class="Symbol">(</a><a id="10372" href="Categories.Morphism.Reasoning.Core.html#2048" class="Function">pullˡ</a> <a id="10378" class="Symbol">(</a><a id="10379" href="Categories.FreeObjects.Free.html#937" class="Field">FreeObject.*-lift</a> <a id="10397" class="Symbol">(</a><a id="10398" href="Monad.Instance.K.html#1182" class="Field">freealgebras</a> <a id="10411" class="Symbol">_)</a> <a id="10414" class="Symbol">(</a><a id="10415" href="Categories.NaturalTransformation.Core.html#783" class="Function">T.η.η</a> <a id="10421" class="Symbol">_)))</a> <a id="10426" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
<a id="10442" href="Categories.NaturalTransformation.Core.html#783" class="Function">T.μ.η</a> <a id="10448" href="Monad.Instance.K.PreElgot.html#6709" class="Bound">X</a> <a id="10450" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="10452" href="Categories.NaturalTransformation.Core.html#783" class="Function">T.η.η</a> <a id="10458" class="Symbol">_</a> <a id="10460" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="10462" href="Monad.Instance.K.PreElgot.html#2394" class="Function">η&#39;</a> <a id="10465" href="Monad.Instance.K.PreElgot.html#6709" class="Bound">X</a> <a id="10495" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="10498" href="Categories.Morphism.Reasoning.Core.html#6851" class="Function">cancelˡ</a> <a id="10506" class="Symbol">(</a><a id="10507" href="Categories.Monad.html#953" class="Field">Monad.identityʳ</a> <a id="10523" href="Monad.PreElgot.html#1214" class="Field">T</a><a id="10524" class="Symbol">)</a> <a id="10526" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
<a id="10542" href="Monad.Instance.K.PreElgot.html#2394" class="Function">η&#39;</a> <a id="10545" href="Monad.Instance.K.PreElgot.html#6709" class="Bound">X</a> <a id="10595" href="Relation.Binary.Reasoning.Syntax.html#12283" class="Function Operator">∎</a>
<a id="10601" href="Monad.Instance.K.PreElgot.html#10601" class="Function">!-unique′</a> <a id="10611" class="Symbol">:</a> <a id="10613" class="Symbol">∀</a> <a id="10615" class="Symbol">{</a><a id="10616" href="Monad.Instance.K.PreElgot.html#10616" class="Bound">A</a> <a id="10618" class="Symbol">:</a> <a id="10620" href="Monad.PreElgot.html#1160" class="Record">PreElgotMonad</a><a id="10633" class="Symbol">}</a> <a id="10635" class="Symbol">(</a><a id="10636" href="Monad.Instance.K.PreElgot.html#10636" class="Bound">f</a> <a id="10638" class="Symbol">:</a> <a id="10640" href="Category.Construction.PreElgotMonads.html#1190" class="Record">PreElgotMonad-Morphism</a> <a id="10663" href="Monad.Instance.K.PreElgot.html#1238" class="Function">preElgot</a> <a id="10672" href="Monad.Instance.K.PreElgot.html#10616" class="Bound">A</a><a id="10673" class="Symbol">)</a> <a id="10675" class="Symbol">→</a> <a id="10677" href="Category.Construction.PreElgotMonads.html#1253" class="Field">PreElgotMonad-Morphism.α</a> <a id="10702" class="Symbol">(</a><a id="10703" href="Monad.Instance.K.PreElgot.html#1456" class="Function">!′</a> <a id="10706" class="Symbol">{</a><a id="10707" class="Argument">A</a> <a id="10709" class="Symbol">=</a> <a id="10711" href="Monad.Instance.K.PreElgot.html#10616" class="Bound">A</a><a id="10712" class="Symbol">})</a> <a id="10715" href="Categories.NaturalTransformation.Equivalence.html#630" class="Function Operator">≃</a> <a id="10717" href="Category.Construction.PreElgotMonads.html#1253" class="Field">PreElgotMonad-Morphism.α</a> <a id="10742" href="Monad.Instance.K.PreElgot.html#10636" class="Bound">f</a>
<a id="10748" href="Monad.Instance.K.PreElgot.html#10601" class="Function">!-unique′</a> <a id="10758" class="Symbol">{</a><a id="10759" href="Monad.Instance.K.PreElgot.html#10759" class="Bound">A</a><a id="10760" class="Symbol">}</a> <a id="10762" href="Monad.Instance.K.PreElgot.html#10762" class="Bound">f</a> <a id="10764" class="Symbol">{</a><a id="10765" href="Monad.Instance.K.PreElgot.html#10765" class="Bound">X</a><a id="10766" class="Symbol">}</a> <a id="10768" class="Symbol">=</a> <a id="10770" href="Relation.Binary.Structures.html#1622" class="Function">sym</a> <a id="10774" class="Symbol">(</a><a id="10775" href="Categories.FreeObjects.Free.html#1017" class="Field">FreeObject.*-uniq</a>
<a id="10825" class="Symbol">(</a><a id="10826" href="Monad.Instance.K.html#1182" class="Field">freealgebras</a> <a id="10839" href="Monad.Instance.K.PreElgot.html#10765" class="Bound">X</a><a id="10840" class="Symbol">)</a>
<a id="10875" class="Symbol">{</a><a id="10876" class="Argument">A</a> <a id="10878" class="Symbol">=</a> <a id="10880" class="Keyword">record</a> <a id="10887" class="Symbol">{</a> <a id="10889" href="Algebra.Elgot.html#16018" class="Field">A</a> <a id="10891" class="Symbol">=</a> <a id="10893" href="Categories.Functor.Core.html#432" class="Function">T.F.F₀</a> <a id="10900" href="Monad.Instance.K.PreElgot.html#10765" class="Bound">X</a> <a id="10902" class="Symbol">;</a> <a id="10904" href="Algebra.Elgot.html#16032" class="Field">algebra</a> <a id="10912" class="Symbol">=</a> <a id="10914" href="Monad.PreElgot.html#815" class="Function">PreElgotMonad.elgotalgebras</a> <a id="10942" href="Monad.Instance.K.PreElgot.html#10759" class="Bound">A</a> <a id="10944" class="Symbol">}}</a>
<a id="10980" class="Symbol">(</a><a id="10981" href="Categories.NaturalTransformation.Core.html#783" class="Function">T.η.η</a> <a id="10987" href="Monad.Instance.K.PreElgot.html#10765" class="Bound">X</a><a id="10988" class="Symbol">)</a>
<a id="11023" class="Symbol">(</a><a id="11024" class="Keyword">record</a> <a id="11031" class="Symbol">{</a> <a id="11033" href="Category.Construction.ElgotAlgebras.html#794" class="Field">h</a> <a id="11035" class="Symbol">=</a> <a id="11037" href="Categories.NaturalTransformation.Core.html#783" class="Function">α.η</a> <a id="11041" href="Monad.Instance.K.PreElgot.html#10765" class="Bound">X</a> <a id="11043" class="Symbol">;</a> <a id="11045" href="Category.Construction.ElgotAlgebras.html#812" class="Field">preserves</a> <a id="11055" class="Symbol">=</a> <a id="11057" href="Category.Construction.PreElgotMonads.html#1490" class="Function">preserves</a> <a id="11067" class="Symbol">_</a> <a id="11069" class="Symbol">})</a>
<a id="11105" href="Category.Construction.PreElgotMonads.html#1344" class="Function">α-η</a><a id="11108" class="Symbol">)</a>
<a id="11116" class="Keyword">where</a>
<a id="11130" class="Keyword">open</a> <a id="11135" href="Category.Construction.PreElgotMonads.html#1190" class="Module">PreElgotMonad-Morphism</a> <a id="11158" href="Monad.Instance.K.PreElgot.html#10762" class="Bound">f</a> <a id="11160" class="Keyword">using</a> <a id="11166" class="Symbol">(</a><a id="11167" href="Category.Construction.PreElgotMonads.html#1253" class="Field">α</a><a id="11168" class="Symbol">;</a> <a id="11170" href="Category.Construction.PreElgotMonads.html#1344" class="Field">α-η</a><a id="11173" class="Symbol">;</a> <a id="11175" href="Category.Construction.PreElgotMonads.html#1490" class="Field">preserves</a><a id="11184" class="Symbol">)</a>
<a id="11194" class="Keyword">open</a> <a id="11199" href="Monad.PreElgot.html#1160" class="Module">PreElgotMonad</a> <a id="11213" href="Monad.Instance.K.PreElgot.html#10759" class="Bound">A</a> <a id="11215" class="Keyword">using</a> <a id="11221" class="Symbol">(</a><a id="11222" href="Monad.PreElgot.html#1214" class="Field">T</a><a id="11223" class="Symbol">)</a>
<a id="11233" class="Keyword">module</a> <a id="11240" href="Monad.Instance.K.PreElgot.html#11240" class="Module">T</a> <a id="11242" class="Symbol">=</a> <a id="11244" href="Categories.Monad.html#454" class="Module">Monad</a> <a id="11250" href="Monad.PreElgot.html#1214" class="Function">T</a>
</pre>
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