bsc-leon-vatthauer/public/Monad.ElgotMonad.md

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## Summary
This file introduces Elgot Monads.

- [X] *Definition 13* Pre-Elgot Monads
- [ ] *Definition 13* strong pre-Elgot
- [X] *Definition 14* Elgot Monads
- [ ] *Definition 14* strong Elgot
- [ ] *Proposition 15* (Strong) Elgot monads are (strong) pre-Elgot

## Code

<pre class="Agda"><a id="752" class="Keyword">module</a> <a id="759" href="Monad.ElgotMonad.html" class="Module">Monad.ElgotMonad</a> <a id="776" class="Symbol">{</a><a id="777" href="Monad.ElgotMonad.html#777" class="Bound">o</a> <a id="779" href="Monad.ElgotMonad.html#779" class="Bound">ℓ</a> <a id="781" href="Monad.ElgotMonad.html#781" class="Bound">e</a><a id="782" class="Symbol">}</a> <a id="784" class="Symbol">(</a><a id="785" href="Monad.ElgotMonad.html#785" class="Bound">ED</a> <a id="788" class="Symbol">:</a> <a id="790" href="Categories.Category.Extensive.Bundle.html#781" class="Record">ExtensiveDistributiveCategory</a> <a id="820" href="Monad.ElgotMonad.html#777" class="Bound">o</a> <a id="822" href="Monad.ElgotMonad.html#779" class="Bound">ℓ</a> <a id="824" href="Monad.ElgotMonad.html#781" class="Bound">e</a><a id="825" class="Symbol">)</a> <a id="827" class="Keyword">where</a>
  <a id="835" class="Keyword">open</a> <a id="840" href="Categories.Category.Extensive.Bundle.html#781" class="Module">ExtensiveDistributiveCategory</a> <a id="870" href="Monad.ElgotMonad.html#785" class="Bound">ED</a> <a id="873" class="Keyword">renaming</a> <a id="882" class="Symbol">(</a><a id="883" href="Categories.Category.Extensive.Bundle.html#859" class="Field">U</a> <a id="885" class="Symbol">to</a> <a id="888" class="Field">C</a><a id="889" class="Symbol">;</a> <a id="891" href="Categories.Category.Core.html#630" class="Function">id</a> <a id="894" class="Symbol">to</a> <a id="897" class="Function">idC</a><a id="900" class="Symbol">)</a>
  <a id="904" class="Keyword">open</a> <a id="909" href="Categories.Category.Core.html#2462" class="Module">HomReasoning</a>
  <a id="924" class="Keyword">open</a> <a id="929" href="Categories.Category.Cocartesian.html#3385" class="Module">Cocartesian</a> <a id="941" class="Symbol">(</a><a id="942" href="Categories.Category.Extensive.html#734" class="Field">Extensive.cocartesian</a> <a id="964" href="Categories.Category.Extensive.Bundle.html#913" class="Field">extensive</a><a id="973" class="Symbol">)</a>
  <a id="977" class="Keyword">open</a> <a id="982" href="Categories.Category.Cartesian.html#727" class="Module">Cartesian</a> <a id="992" class="Symbol">(</a><a id="993" href="Categories.Category.Extensive.Bundle.html#943" class="Field">ExtensiveDistributiveCategory.cartesian</a> <a id="1033" href="Monad.ElgotMonad.html#785" class="Bound">ED</a><a id="1035" class="Symbol">)</a>
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  <a id="1081" class="Keyword">open</a> <a id="1086" href="Categories.Morphism.Reasoning.html" class="Module">MR</a> <a id="1089" href="Monad.ElgotMonad.html#888" class="Field">C</a>
  <a id="1093" class="Keyword">open</a> <a id="1098" href="Categories.Category.Core.html#1530" class="Module">Equiv</a>
</pre>
### *Definition 13*: Pre-Elgot Monads

<pre class="Agda">  <a id="1158" class="Keyword">record</a> <a id="IsPreElgot"></a><a id="1165" href="Monad.ElgotMonad.html#1165" class="Record">IsPreElgot</a> <a id="1176" class="Symbol">(</a><a id="1177" href="Monad.ElgotMonad.html#1177" class="Bound">T</a> <a id="1179" class="Symbol">:</a> <a id="1181" href="Categories.Monad.html#454" class="Record">Monad</a> <a id="1187" href="Monad.ElgotMonad.html#888" class="Field">C</a><a id="1188" class="Symbol">)</a> <a id="1190" class="Symbol">:</a> <a id="1192" href="Agda.Primitive.html#320" class="Primitive">Set</a> <a id="1196" class="Symbol">(</a><a id="1197" href="Monad.ElgotMonad.html#777" class="Bound">o</a> <a id="1199" href="Agda.Primitive.html#804" class="Primitive Operator">⊔</a> <a id="1201" href="Monad.ElgotMonad.html#779" class="Bound">ℓ</a> <a id="1203" href="Agda.Primitive.html#804" class="Primitive Operator">⊔</a> <a id="1205" href="Monad.ElgotMonad.html#781" class="Bound">e</a><a id="1206" class="Symbol">)</a> <a id="1208" class="Keyword">where</a>
    <a id="1218" class="Keyword">open</a> <a id="1223" href="Categories.Monad.html#454" class="Module">Monad</a> <a id="1229" href="Monad.ElgotMonad.html#1177" class="Bound">T</a>
    <a id="1235" class="Keyword">open</a> <a id="1240" href="Categories.Functor.Core.html#248" class="Module">Functor</a> <a id="1248" href="Categories.Monad.html#525" class="Function">F</a> <a id="1250" class="Keyword">renaming</a> <a id="1259" class="Symbol">(</a><a id="1260" href="Categories.Functor.Core.html#432" class="Field">F₀</a> <a id="1263" class="Symbol">to</a> <a id="1266" class="Field">T₀</a><a id="1268" class="Symbol">;</a> <a id="1270" href="Categories.Functor.Core.html#455" class="Field">F₁</a> <a id="1273" class="Symbol">to</a> <a id="1276" class="Field">T₁</a><a id="1278" class="Symbol">)</a>

    <a id="1285" class="Comment">-- every TX needs to be equipped with an elgot algebra structure</a>
    <a id="1354" class="Keyword">field</a>
      <a id="IsPreElgot.elgotalgebras"></a><a id="1366" href="Monad.ElgotMonad.html#1366" class="Field">elgotalgebras</a> <a id="1380" class="Symbol">:</a> <a id="1382" class="Symbol">∀</a> <a id="1384" class="Symbol">{</a><a id="1385" href="Monad.ElgotMonad.html#1385" class="Bound">X</a><a id="1386" class="Symbol">}</a> <a id="1388" class="Symbol">→</a> <a id="1390" href="ElgotAlgebra.html#2093" class="Record">Elgot-Algebra-on</a> <a id="1407" href="Monad.ElgotMonad.html#785" class="Bound">ED</a> <a id="1410" class="Symbol">(</a><a id="1411" href="Monad.ElgotMonad.html#1266" class="Function">T₀</a> <a id="1414" href="Monad.ElgotMonad.html#1385" class="Bound">X</a><a id="1415" class="Symbol">)</a>

    <a id="1422" class="Keyword">module</a> <a id="IsPreElgot.elgotalgebras"></a><a id="1429" href="Monad.ElgotMonad.html#1429" class="Module">elgotalgebras</a> <a id="1443" class="Symbol">{</a><a id="1444" href="Monad.ElgotMonad.html#1444" class="Bound">X</a><a id="1445" class="Symbol">}</a> <a id="1447" class="Symbol">=</a> <a id="1449" href="ElgotAlgebra.html#2093" class="Module">Elgot-Algebra-on</a> <a id="1466" class="Symbol">(</a><a id="1467" href="Monad.ElgotMonad.html#1366" class="Field">elgotalgebras</a> <a id="1481" class="Symbol">{</a><a id="1482" href="Monad.ElgotMonad.html#1444" class="Bound">X</a><a id="1483" class="Symbol">})</a>

    <a id="1491" class="Comment">-- with the following associativity</a>
    <a id="1531" class="Keyword">field</a>
      <a id="IsPreElgot.assoc"></a><a id="1543" href="Monad.ElgotMonad.html#1543" class="Field">assoc</a> <a id="1549" class="Symbol">:</a> <a id="1551" class="Symbol">∀</a> <a id="1553" class="Symbol">{</a><a id="1554" href="Monad.ElgotMonad.html#1554" class="Bound">X</a> <a id="1556" href="Monad.ElgotMonad.html#1556" class="Bound">Y</a> <a id="1558" href="Monad.ElgotMonad.html#1558" class="Bound">Z</a><a id="1559" class="Symbol">}</a> <a id="1561" class="Symbol">(</a><a id="1562" href="Monad.ElgotMonad.html#1562" class="Bound">f</a> <a id="1564" class="Symbol">:</a> <a id="1566" href="Monad.ElgotMonad.html#1558" class="Bound">Z</a> <a id="1568" href="Categories.Category.Core.html#575" class="Function Operator">⇒</a> <a id="1570" href="Monad.ElgotMonad.html#1266" class="Function">T₀</a> <a id="1573" href="Monad.ElgotMonad.html#1554" class="Bound">X</a> <a id="1575" href="Categories.Category.Cocartesian.html#1549" class="Function Operator">+</a> <a id="1577" href="Monad.ElgotMonad.html#1558" class="Bound">Z</a><a id="1578" class="Symbol">)</a> <a id="1580" class="Symbol">(</a><a id="1581" href="Monad.ElgotMonad.html#1581" class="Bound">h</a> <a id="1583" class="Symbol">:</a> <a id="1585" href="Monad.ElgotMonad.html#1554" class="Bound">X</a> <a id="1587" href="Categories.Category.Core.html#575" class="Function Operator">⇒</a> <a id="1589" href="Monad.ElgotMonad.html#1266" class="Function">T₀</a> <a id="1592" href="Monad.ElgotMonad.html#1556" class="Bound">Y</a><a id="1593" class="Symbol">)</a> 
        <a id="1604" class="Symbol">→</a> <a id="1606" href="ElgotAlgebra.html#2186" class="Function Operator">elgotalgebras._#</a> <a id="1623" class="Symbol">(((</a><a id="1626" href="Categories.NaturalTransformation.Core.html#783" class="Function">μ.η</a> <a id="1630" class="Symbol">_</a> <a id="1632" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="1634" href="Monad.ElgotMonad.html#1276" class="Function">T₁</a> <a id="1637" href="Monad.ElgotMonad.html#1581" class="Bound">h</a><a id="1638" class="Symbol">)</a> <a id="1640" href="Categories.Category.Cocartesian.html#2133" class="Function Operator">+₁</a> <a id="1643" href="Monad.ElgotMonad.html#897" class="Function">idC</a><a id="1646" class="Symbol">)</a> <a id="1648" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="1650" href="Monad.ElgotMonad.html#1562" class="Bound">f</a><a id="1651" class="Symbol">)</a> <a id="1653" href="Categories.Category.Core.html#595" class="Function Operator">≈</a> <a id="1655" class="Symbol">(</a><a id="1656" href="Categories.NaturalTransformation.Core.html#783" class="Function">μ.η</a> <a id="1660" class="Symbol">_</a> <a id="1662" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="1664" href="Monad.ElgotMonad.html#1276" class="Function">T₁</a> <a id="1667" href="Monad.ElgotMonad.html#1581" class="Bound">h</a><a id="1668" class="Symbol">)</a> <a id="1670" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="1672" class="Symbol">(</a><a id="1673" href="ElgotAlgebra.html#2186" class="Function Operator">elgotalgebras._#</a> <a id="1690" class="Symbol">{</a><a id="1691" href="Monad.ElgotMonad.html#1554" class="Bound">X</a><a id="1692" class="Symbol">})</a> <a id="1695" href="Monad.ElgotMonad.html#1562" class="Bound">f</a>
  
  <a id="1702" class="Keyword">record</a> <a id="PreElgotMonad"></a><a id="1709" href="Monad.ElgotMonad.html#1709" class="Record">PreElgotMonad</a> <a id="1723" class="Symbol">:</a> <a id="1725" href="Agda.Primitive.html#320" class="Primitive">Set</a> <a id="1729" class="Symbol">(</a><a id="1730" href="Monad.ElgotMonad.html#777" class="Bound">o</a> <a id="1732" href="Agda.Primitive.html#804" class="Primitive Operator">⊔</a> <a id="1734" href="Monad.ElgotMonad.html#779" class="Bound">ℓ</a> <a id="1736" href="Agda.Primitive.html#804" class="Primitive Operator">⊔</a> <a id="1738" href="Monad.ElgotMonad.html#781" class="Bound">e</a><a id="1739" class="Symbol">)</a> <a id="1741" class="Keyword">where</a>
    <a id="1751" class="Keyword">field</a>
      <a id="PreElgotMonad.T"></a><a id="1763" href="Monad.ElgotMonad.html#1763" class="Field">T</a> <a id="1765" class="Symbol">:</a> <a id="1767" href="Categories.Monad.html#454" class="Record">Monad</a> <a id="1773" href="Monad.ElgotMonad.html#888" class="Field">C</a>
      <a id="PreElgotMonad.isPreElgot"></a><a id="1781" href="Monad.ElgotMonad.html#1781" class="Field">isPreElgot</a> <a id="1792" class="Symbol">:</a> <a id="1794" href="Monad.ElgotMonad.html#1165" class="Record">IsPreElgot</a> <a id="1805" href="Monad.ElgotMonad.html#1763" class="Field">T</a>

    <a id="1812" class="Keyword">open</a> <a id="1817" href="Monad.ElgotMonad.html#1165" class="Module">IsPreElgot</a> <a id="1828" href="Monad.ElgotMonad.html#1781" class="Field">isPreElgot</a> <a id="1839" class="Keyword">public</a>
</pre>### *Definition 14*: Elgot Monads

<pre class="Agda">  <a id="1895" class="Keyword">record</a> <a id="IsElgot"></a><a id="1902" href="Monad.ElgotMonad.html#1902" class="Record">IsElgot</a> <a id="1910" class="Symbol">(</a><a id="1911" href="Monad.ElgotMonad.html#1911" class="Bound">T</a> <a id="1913" class="Symbol">:</a> <a id="1915" href="Categories.Monad.html#454" class="Record">Monad</a> <a id="1921" href="Monad.ElgotMonad.html#888" class="Field">C</a><a id="1922" class="Symbol">)</a> <a id="1924" class="Symbol">:</a> <a id="1926" href="Agda.Primitive.html#320" class="Primitive">Set</a> <a id="1930" class="Symbol">(</a><a id="1931" href="Monad.ElgotMonad.html#777" class="Bound">o</a> <a id="1933" href="Agda.Primitive.html#804" class="Primitive Operator">⊔</a> <a id="1935" href="Monad.ElgotMonad.html#779" class="Bound">ℓ</a> <a id="1937" href="Agda.Primitive.html#804" class="Primitive Operator">⊔</a> <a id="1939" href="Monad.ElgotMonad.html#781" class="Bound">e</a><a id="1940" class="Symbol">)</a> <a id="1942" class="Keyword">where</a>
    <a id="1952" class="Keyword">open</a> <a id="1957" href="Categories.Monad.html#454" class="Module">Monad</a> <a id="1963" href="Monad.ElgotMonad.html#1911" class="Bound">T</a>
    <a id="1969" class="Keyword">open</a> <a id="1974" href="Categories.Functor.Core.html#248" class="Module">Functor</a> <a id="1982" href="Categories.Monad.html#525" class="Function">F</a> <a id="1984" class="Keyword">renaming</a> <a id="1993" class="Symbol">(</a><a id="1994" href="Categories.Functor.Core.html#432" class="Field">F₀</a> <a id="1997" class="Symbol">to</a> <a id="2000" class="Field">T₀</a><a id="2002" class="Symbol">;</a> <a id="2004" href="Categories.Functor.Core.html#455" class="Field">F₁</a> <a id="2007" class="Symbol">to</a> <a id="2010" class="Field">T₁</a><a id="2012" class="Symbol">)</a>

    <a id="2019" class="Comment">-- iteration operator</a>
    <a id="2045" class="Keyword">field</a>
      <a id="IsElgot._†"></a><a id="2057" href="Monad.ElgotMonad.html#2057" class="Field Operator">_†</a> <a id="2060" class="Symbol">:</a> <a id="2062" class="Symbol">∀</a> <a id="2064" class="Symbol">{</a><a id="2065" href="Monad.ElgotMonad.html#2065" class="Bound">X</a> <a id="2067" href="Monad.ElgotMonad.html#2067" class="Bound">Y</a><a id="2068" class="Symbol">}</a> <a id="2070" class="Symbol">→</a> <a id="2072" href="Monad.ElgotMonad.html#2065" class="Bound">X</a> <a id="2074" href="Categories.Category.Core.html#575" class="Function Operator">⇒</a> <a id="2076" href="Monad.ElgotMonad.html#2000" class="Function">T₀</a> <a id="2079" class="Symbol">(</a><a id="2080" href="Monad.ElgotMonad.html#2067" class="Bound">Y</a> <a id="2082" href="Categories.Category.Cocartesian.html#1549" class="Function Operator">+</a> <a id="2084" href="Monad.ElgotMonad.html#2065" class="Bound">X</a><a id="2085" class="Symbol">)</a> <a id="2087" class="Symbol">→</a> <a id="2089" href="Monad.ElgotMonad.html#2065" class="Bound">X</a> <a id="2091" href="Categories.Category.Core.html#575" class="Function Operator">⇒</a> <a id="2093" href="Monad.ElgotMonad.html#2000" class="Function">T₀</a> <a id="2096" href="Monad.ElgotMonad.html#2067" class="Bound">Y</a>
      <a id="IsElgot.†-resp-≈"></a><a id="2104" href="Monad.ElgotMonad.html#2104" class="Field">†-resp-≈</a> <a id="2113" class="Symbol">:</a> <a id="2115" class="Symbol">∀</a> <a id="2117" class="Symbol">{</a><a id="2118" href="Monad.ElgotMonad.html#2118" class="Bound">X</a> <a id="2120" href="Monad.ElgotMonad.html#2120" class="Bound">Y</a><a id="2121" class="Symbol">}</a> <a id="2123" class="Symbol">{</a><a id="2124" href="Monad.ElgotMonad.html#2124" class="Bound">f</a> <a id="2126" href="Monad.ElgotMonad.html#2126" class="Bound">g</a> <a id="2128" class="Symbol">:</a> <a id="2130" href="Monad.ElgotMonad.html#2118" class="Bound">X</a> <a id="2132" href="Categories.Category.Core.html#575" class="Function Operator">⇒</a> <a id="2134" href="Monad.ElgotMonad.html#2000" class="Function">T₀</a> <a id="2137" class="Symbol">(</a><a id="2138" href="Monad.ElgotMonad.html#2120" class="Bound">Y</a> <a id="2140" href="Categories.Category.Cocartesian.html#1549" class="Function Operator">+</a> <a id="2142" href="Monad.ElgotMonad.html#2118" class="Bound">X</a><a id="2143" class="Symbol">)}</a> <a id="2146" class="Symbol">→</a> <a id="2148" href="Monad.ElgotMonad.html#2124" class="Bound">f</a> <a id="2150" href="Categories.Category.Core.html#595" class="Function Operator">≈</a> <a id="2152" href="Monad.ElgotMonad.html#2126" class="Bound">g</a> <a id="2154" class="Symbol">→</a> <a id="2156" href="Monad.ElgotMonad.html#2124" class="Bound">f</a> <a id="2158" href="Monad.ElgotMonad.html#2057" class="Field Operator">†</a> <a id="2160" href="Categories.Category.Core.html#595" class="Function Operator">≈</a> <a id="2162" href="Monad.ElgotMonad.html#2126" class="Bound">g</a> <a id="2164" href="Monad.ElgotMonad.html#2057" class="Field Operator">†</a>

    <a id="2171" class="Comment">-- laws</a>
    <a id="2183" class="Keyword">field</a>
      <a id="IsElgot.Fixpoint"></a><a id="2195" href="Monad.ElgotMonad.html#2195" class="Field">Fixpoint</a> <a id="2204" class="Symbol">:</a> <a id="2206" class="Symbol">∀</a> <a id="2208" class="Symbol">{</a><a id="2209" href="Monad.ElgotMonad.html#2209" class="Bound">X</a> <a id="2211" href="Monad.ElgotMonad.html#2211" class="Bound">Y</a><a id="2212" class="Symbol">}</a> <a id="2214" class="Symbol">{</a><a id="2215" href="Monad.ElgotMonad.html#2215" class="Bound">f</a> <a id="2217" class="Symbol">:</a> <a id="2219" href="Monad.ElgotMonad.html#2209" class="Bound">X</a> <a id="2221" href="Categories.Category.Core.html#575" class="Function Operator">⇒</a> <a id="2223" href="Monad.ElgotMonad.html#2000" class="Function">T₀</a> <a id="2226" class="Symbol">(</a><a id="2227" href="Monad.ElgotMonad.html#2211" class="Bound">Y</a> <a id="2229" href="Categories.Category.Cocartesian.html#1549" class="Function Operator">+</a> <a id="2231" href="Monad.ElgotMonad.html#2209" class="Bound">X</a><a id="2232" class="Symbol">)}</a> 
        <a id="2244" class="Symbol">→</a> <a id="2246" href="Monad.ElgotMonad.html#2215" class="Bound">f</a> <a id="2248" href="Monad.ElgotMonad.html#2057" class="Field Operator">†</a> <a id="2250" href="Categories.Category.Core.html#595" class="Function Operator">≈</a> <a id="2252" class="Symbol">(</a><a id="2253" href="Categories.NaturalTransformation.Core.html#783" class="Function">μ.η</a> <a id="2257" class="Symbol">_</a> <a id="2259" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="2261" href="Monad.ElgotMonad.html#2010" class="Function">T₁</a> <a id="2264" href="Categories.Object.Coproduct.html#532" class="Function Operator">[</a> <a id="2266" href="Categories.NaturalTransformation.Core.html#783" class="Function">η.η</a> <a id="2270" class="Symbol">_</a> <a id="2272" href="Categories.Object.Coproduct.html#532" class="Function Operator">,</a> <a id="2274" href="Monad.ElgotMonad.html#2215" class="Bound">f</a> <a id="2276" href="Monad.ElgotMonad.html#2057" class="Field Operator">†</a> <a id="2278" href="Categories.Object.Coproduct.html#532" class="Function Operator">]</a><a id="2279" class="Symbol">)</a> <a id="2281" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="2283" href="Monad.ElgotMonad.html#2215" class="Bound">f</a>
      <a id="IsElgot.Naturality"></a><a id="2291" href="Monad.ElgotMonad.html#2291" class="Field">Naturality</a> <a id="2302" class="Symbol">:</a> <a id="2304" class="Symbol">∀</a> <a id="2306" class="Symbol">{</a><a id="2307" href="Monad.ElgotMonad.html#2307" class="Bound">X</a> <a id="2309" href="Monad.ElgotMonad.html#2309" class="Bound">Y</a> <a id="2311" href="Monad.ElgotMonad.html#2311" class="Bound">Z</a><a id="2312" class="Symbol">}</a> <a id="2314" class="Symbol">{</a><a id="2315" href="Monad.ElgotMonad.html#2315" class="Bound">f</a> <a id="2317" class="Symbol">:</a> <a id="2319" href="Monad.ElgotMonad.html#2307" class="Bound">X</a> <a id="2321" href="Categories.Category.Core.html#575" class="Function Operator">⇒</a> <a id="2323" href="Monad.ElgotMonad.html#2000" class="Function">T₀</a> <a id="2326" class="Symbol">(</a><a id="2327" href="Monad.ElgotMonad.html#2309" class="Bound">Y</a> <a id="2329" href="Categories.Category.Cocartesian.html#1549" class="Function Operator">+</a> <a id="2331" href="Monad.ElgotMonad.html#2307" class="Bound">X</a><a id="2332" class="Symbol">)}</a> <a id="2335" class="Symbol">{</a><a id="2336" href="Monad.ElgotMonad.html#2336" class="Bound">g</a> <a id="2338" class="Symbol">:</a> <a id="2340" href="Monad.ElgotMonad.html#2309" class="Bound">Y</a> <a id="2342" href="Categories.Category.Core.html#575" class="Function Operator">⇒</a> <a id="2344" href="Monad.ElgotMonad.html#2000" class="Function">T₀</a> <a id="2347" href="Monad.ElgotMonad.html#2311" class="Bound">Z</a><a id="2348" class="Symbol">}</a> 
        <a id="2359" class="Symbol">→</a> <a id="2361" class="Symbol">(</a><a id="2362" href="Categories.NaturalTransformation.Core.html#783" class="Function">μ.η</a> <a id="2366" class="Symbol">_</a> <a id="2368" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="2370" href="Monad.ElgotMonad.html#2010" class="Function">T₁</a> <a id="2373" href="Monad.ElgotMonad.html#2336" class="Bound">g</a><a id="2374" class="Symbol">)</a> <a id="2376" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="2378" href="Monad.ElgotMonad.html#2315" class="Bound">f</a> <a id="2380" href="Monad.ElgotMonad.html#2057" class="Field Operator">†</a> <a id="2382" href="Categories.Category.Core.html#595" class="Function Operator">≈</a> <a id="2384" class="Symbol">((</a><a id="2386" href="Categories.NaturalTransformation.Core.html#783" class="Function">μ.η</a> <a id="2390" class="Symbol">_</a> <a id="2392" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="2394" href="Monad.ElgotMonad.html#2010" class="Function">T₁</a> <a id="2397" href="Categories.Object.Coproduct.html#532" class="Function Operator">[</a> <a id="2399" class="Symbol">(</a><a id="2400" href="Monad.ElgotMonad.html#2010" class="Function">T₁</a> <a id="2403" href="Categories.Object.Coproduct.html#492" class="Function">i₁</a><a id="2405" class="Symbol">)</a> <a id="2407" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="2409" href="Monad.ElgotMonad.html#2336" class="Bound">g</a> <a id="2411" href="Categories.Object.Coproduct.html#532" class="Function Operator">,</a> <a id="2413" href="Categories.NaturalTransformation.Core.html#783" class="Function">η.η</a> <a id="2417" class="Symbol">_</a> <a id="2419" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="2421" href="Categories.Object.Coproduct.html#512" class="Function">i₂</a> <a id="2424" href="Categories.Object.Coproduct.html#532" class="Function Operator">]</a><a id="2425" class="Symbol">)</a> <a id="2427" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="2429" href="Monad.ElgotMonad.html#2315" class="Bound">f</a><a id="2430" class="Symbol">)</a><a id="2431" href="Monad.ElgotMonad.html#2057" class="Field Operator">†</a>
      <a id="IsElgot.Codiagonal"></a><a id="2439" href="Monad.ElgotMonad.html#2439" class="Field">Codiagonal</a> <a id="2450" class="Symbol">:</a> <a id="2452" class="Symbol">∀</a> <a id="2454" class="Symbol">{</a><a id="2455" href="Monad.ElgotMonad.html#2455" class="Bound">X</a> <a id="2457" href="Monad.ElgotMonad.html#2457" class="Bound">Y</a><a id="2458" class="Symbol">}</a> <a id="2460" class="Symbol">{</a><a id="2461" href="Monad.ElgotMonad.html#2461" class="Bound">f</a> <a id="2463" class="Symbol">:</a> <a id="2465" href="Monad.ElgotMonad.html#2455" class="Bound">X</a> <a id="2467" href="Categories.Category.Core.html#575" class="Function Operator">⇒</a> <a id="2469" href="Monad.ElgotMonad.html#2000" class="Function">T₀</a> <a id="2472" class="Symbol">((</a><a id="2474" href="Monad.ElgotMonad.html#2457" class="Bound">Y</a> <a id="2476" href="Categories.Category.Cocartesian.html#1549" class="Function Operator">+</a> <a id="2478" href="Monad.ElgotMonad.html#2455" class="Bound">X</a><a id="2479" class="Symbol">)</a> <a id="2481" href="Categories.Category.Cocartesian.html#1549" class="Function Operator">+</a> <a id="2483" href="Monad.ElgotMonad.html#2455" class="Bound">X</a><a id="2484" class="Symbol">)}</a> 
        <a id="2496" class="Symbol">→</a> <a id="2498" class="Symbol">(</a><a id="2499" href="Monad.ElgotMonad.html#2010" class="Function">T₁</a> <a id="2502" href="Categories.Object.Coproduct.html#532" class="Function Operator">[</a> <a id="2504" href="Monad.ElgotMonad.html#897" class="Function">idC</a> <a id="2508" href="Categories.Object.Coproduct.html#532" class="Function Operator">,</a> <a id="2510" href="Categories.Object.Coproduct.html#512" class="Function">i₂</a> <a id="2513" href="Categories.Object.Coproduct.html#532" class="Function Operator">]</a> <a id="2515" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="2517" href="Monad.ElgotMonad.html#2461" class="Bound">f</a> <a id="2519" class="Symbol">)</a><a id="2520" href="Monad.ElgotMonad.html#2057" class="Field Operator">†</a> <a id="2522" href="Categories.Category.Core.html#595" class="Function Operator">≈</a> <a id="2524" href="Monad.ElgotMonad.html#2461" class="Bound">f</a> <a id="2526" href="Monad.ElgotMonad.html#2057" class="Field Operator">†</a> <a id="2528" href="Monad.ElgotMonad.html#2057" class="Field Operator">†</a>
      <a id="IsElgot.Uniformity"></a><a id="2536" href="Monad.ElgotMonad.html#2536" class="Field">Uniformity</a> <a id="2547" class="Symbol">:</a> <a id="2549" class="Symbol">∀</a> <a id="2551" class="Symbol">{</a><a id="2552" href="Monad.ElgotMonad.html#2552" class="Bound">X</a> <a id="2554" href="Monad.ElgotMonad.html#2554" class="Bound">Y</a> <a id="2556" href="Monad.ElgotMonad.html#2556" class="Bound">Z</a><a id="2557" class="Symbol">}</a> <a id="2559" class="Symbol">{</a><a id="2560" href="Monad.ElgotMonad.html#2560" class="Bound">f</a> <a id="2562" class="Symbol">:</a> <a id="2564" href="Monad.ElgotMonad.html#2552" class="Bound">X</a> <a id="2566" href="Categories.Category.Core.html#575" class="Function Operator">⇒</a> <a id="2568" href="Monad.ElgotMonad.html#2000" class="Function">T₀</a> <a id="2571" class="Symbol">(</a><a id="2572" href="Monad.ElgotMonad.html#2554" class="Bound">Y</a> <a id="2574" href="Categories.Category.Cocartesian.html#1549" class="Function Operator">+</a> <a id="2576" href="Monad.ElgotMonad.html#2552" class="Bound">X</a><a id="2577" class="Symbol">)}</a> <a id="2580" class="Symbol">{</a><a id="2581" href="Monad.ElgotMonad.html#2581" class="Bound">g</a> <a id="2583" class="Symbol">:</a> <a id="2585" href="Monad.ElgotMonad.html#2556" class="Bound">Z</a> <a id="2587" href="Categories.Category.Core.html#575" class="Function Operator">⇒</a> <a id="2589" href="Monad.ElgotMonad.html#2000" class="Function">T₀</a> <a id="2592" class="Symbol">(</a><a id="2593" href="Monad.ElgotMonad.html#2554" class="Bound">Y</a> <a id="2595" href="Categories.Category.Cocartesian.html#1549" class="Function Operator">+</a> <a id="2597" href="Monad.ElgotMonad.html#2556" class="Bound">Z</a><a id="2598" class="Symbol">)}</a> <a id="2601" class="Symbol">{</a><a id="2602" href="Monad.ElgotMonad.html#2602" class="Bound">h</a> <a id="2604" class="Symbol">:</a> <a id="2606" href="Monad.ElgotMonad.html#2556" class="Bound">Z</a> <a id="2608" href="Categories.Category.Core.html#575" class="Function Operator">⇒</a> <a id="2610" href="Monad.ElgotMonad.html#2552" class="Bound">X</a><a id="2611" class="Symbol">}</a> 
        <a id="2622" class="Symbol">→</a> <a id="2624" href="Monad.ElgotMonad.html#2560" class="Bound">f</a> <a id="2626" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="2628" href="Monad.ElgotMonad.html#2602" class="Bound">h</a> <a id="2630" href="Categories.Category.Core.html#595" class="Function Operator">≈</a> <a id="2632" class="Symbol">(</a><a id="2633" href="Monad.ElgotMonad.html#2010" class="Function">T₁</a> <a id="2636" class="Symbol">(</a><a id="2637" href="Monad.ElgotMonad.html#897" class="Function">idC</a> <a id="2641" href="Categories.Category.Cocartesian.html#2133" class="Function Operator">+₁</a> <a id="2644" href="Monad.ElgotMonad.html#2602" class="Bound">h</a><a id="2645" class="Symbol">))</a> <a id="2648" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="2650" href="Monad.ElgotMonad.html#2581" class="Bound">g</a> <a id="2652" class="Symbol">→</a> <a id="2654" href="Monad.ElgotMonad.html#2560" class="Bound">f</a> <a id="2656" href="Monad.ElgotMonad.html#2057" class="Field Operator">†</a> <a id="2658" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="2660" href="Monad.ElgotMonad.html#2602" class="Bound">h</a> <a id="2662" href="Categories.Category.Core.html#595" class="Function Operator">≈</a> <a id="2664" href="Monad.ElgotMonad.html#2581" class="Bound">g</a> <a id="2666" href="Monad.ElgotMonad.html#2057" class="Field Operator">†</a>

  <a id="2671" class="Keyword">record</a> <a id="ElgotMonad"></a><a id="2678" href="Monad.ElgotMonad.html#2678" class="Record">ElgotMonad</a> <a id="2689" class="Symbol">:</a> <a id="2691" href="Agda.Primitive.html#320" class="Primitive">Set</a> <a id="2695" class="Symbol">(</a><a id="2696" href="Monad.ElgotMonad.html#777" class="Bound">o</a> <a id="2698" href="Agda.Primitive.html#804" class="Primitive Operator">⊔</a> <a id="2700" href="Monad.ElgotMonad.html#779" class="Bound">ℓ</a> <a id="2702" href="Agda.Primitive.html#804" class="Primitive Operator">⊔</a> <a id="2704" href="Monad.ElgotMonad.html#781" class="Bound">e</a><a id="2705" class="Symbol">)</a> <a id="2707" class="Keyword">where</a>
    <a id="2717" class="Keyword">field</a>
      <a id="ElgotMonad.T"></a><a id="2729" href="Monad.ElgotMonad.html#2729" class="Field">T</a> <a id="2731" class="Symbol">:</a> <a id="2733" href="Categories.Monad.html#454" class="Record">Monad</a> <a id="2739" href="Monad.ElgotMonad.html#888" class="Field">C</a>
      <a id="ElgotMonad.isElgot"></a><a id="2747" href="Monad.ElgotMonad.html#2747" class="Field">isElgot</a> <a id="2755" class="Symbol">:</a> <a id="2757" href="Monad.ElgotMonad.html#1902" class="Record">IsElgot</a> <a id="2765" href="Monad.ElgotMonad.html#2729" class="Field">T</a>

    <a id="2772" class="Keyword">open</a> <a id="2777" href="Monad.ElgotMonad.html#1902" class="Module">IsElgot</a> <a id="2785" href="Monad.ElgotMonad.html#2747" class="Field">isElgot</a> <a id="2793" class="Keyword">public</a>
</pre>
### *Proposition 15*: (Strong) Elgot monads are (strong) pre-Elgot

<pre class="Agda">  <a id="2883" class="Comment">-- elgot monads are pre-elgot</a>
  <a id="Elgot⇒PreElgot"></a><a id="2915" href="Monad.ElgotMonad.html#2915" class="Function">Elgot⇒PreElgot</a> <a id="2930" class="Symbol">:</a> <a id="2932" href="Monad.ElgotMonad.html#2678" class="Record">ElgotMonad</a> <a id="2943" class="Symbol">→</a> <a id="2945" href="Monad.ElgotMonad.html#1709" class="Record">PreElgotMonad</a>
  <a id="2961" href="Monad.ElgotMonad.html#2915" class="Function">Elgot⇒PreElgot</a> <a id="2976" href="Monad.ElgotMonad.html#2976" class="Bound">EM</a> <a id="2979" class="Symbol">=</a> <a id="2981" class="Keyword">record</a> 
    <a id="2993" class="Symbol">{</a> <a id="2995" href="Monad.ElgotMonad.html#1763" class="Field">T</a> <a id="2997" class="Symbol">=</a> <a id="2999" href="Monad.ElgotMonad.html#2729" class="Field">T</a> 
    <a id="3006" class="Symbol">;</a> <a id="3008" href="Monad.ElgotMonad.html#1781" class="Field">isPreElgot</a> <a id="3019" class="Symbol">=</a> <a id="3021" class="Keyword">record</a> 
      <a id="3035" class="Symbol">{</a> <a id="3037" href="Monad.ElgotMonad.html#1366" class="Field">elgotalgebras</a> <a id="3051" class="Symbol">=</a> <a id="3053" class="Symbol">λ</a> <a id="3055" class="Symbol">{</a><a id="3056" href="Monad.ElgotMonad.html#3056" class="Bound">X</a><a id="3057" class="Symbol">}</a> <a id="3059" class="Symbol">→</a> <a id="3061" class="Keyword">record</a>
        <a id="3076" class="Symbol">{</a> <a id="3078" href="ElgotAlgebra.html#2186" class="Field Operator">_#</a> <a id="3081" class="Symbol">=</a> <a id="3083" class="Symbol">λ</a> <a id="3085" href="Monad.ElgotMonad.html#3085" class="Bound">f</a> <a id="3087" class="Symbol">→</a> <a id="3089" class="Symbol">(</a><a id="3090" href="Categories.Object.Coproduct.html#532" class="Function Operator">[</a> <a id="3092" href="Monad.ElgotMonad.html#7809" class="Function">T₁</a> <a id="3095" href="Categories.Object.Coproduct.html#492" class="Function">i₁</a> <a id="3098" href="Categories.Object.Coproduct.html#532" class="Function Operator">,</a> <a id="3100" href="Categories.NaturalTransformation.Core.html#783" class="Function">η.η</a> <a id="3104" class="Symbol">_</a> <a id="3106" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="3108" href="Categories.Object.Coproduct.html#512" class="Function">i₂</a> <a id="3111" href="Categories.Object.Coproduct.html#532" class="Function Operator">]</a> <a id="3113" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="3115" href="Monad.ElgotMonad.html#3085" class="Bound">f</a><a id="3116" class="Symbol">)</a> <a id="3118" href="Monad.ElgotMonad.html#2057" class="Function Operator">†</a>
        <a id="3128" class="Symbol">;</a> <a id="3130" href="ElgotAlgebra.html#2259" class="Field">#-Fixpoint</a> <a id="3141" class="Symbol">=</a> <a id="3143" class="Symbol">λ</a> <a id="3145" class="Symbol">{</a><a id="3146" href="Monad.ElgotMonad.html#3146" class="Bound">Y</a><a id="3147" class="Symbol">}</a> <a id="3149" class="Symbol">{</a><a id="3150" href="Monad.ElgotMonad.html#3150" class="Bound">f</a><a id="3151" class="Symbol">}</a> <a id="3153" class="Symbol">→</a> <a id="3155" href="Relation.Binary.Reasoning.Base.Single.html#1925" class="Function Operator">begin</a> 
          <a id="3172" class="Symbol">(</a><a id="3173" href="Categories.Object.Coproduct.html#532" class="Function Operator">[</a> <a id="3175" href="Monad.ElgotMonad.html#7809" class="Function">T₁</a> <a id="3178" href="Categories.Object.Coproduct.html#492" class="Function">i₁</a> <a id="3181" href="Categories.Object.Coproduct.html#532" class="Function Operator">,</a> <a id="3183" href="Categories.NaturalTransformation.Core.html#783" class="Function">η.η</a> <a id="3187" class="Symbol">_</a> <a id="3189" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="3191" href="Categories.Object.Coproduct.html#512" class="Function">i₂</a> <a id="3194" href="Categories.Object.Coproduct.html#532" class="Function Operator">]</a> <a id="3196" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="3198" href="Monad.ElgotMonad.html#3150" class="Bound">f</a><a id="3199" class="Symbol">)</a> <a id="3201" href="Monad.ElgotMonad.html#2057" class="Function Operator">†</a>                                                         <a id="3259" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">≈⟨</a> <a id="3262" href="Monad.ElgotMonad.html#2195" class="Function">Fixpoint</a> <a id="3271" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">⟩</a> 
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            <a id="5250" href="Monad.ElgotMonad.html#7809" class="Function">T₁</a> <a id="5253" class="Symbol">(</a><a id="5254" href="Monad.ElgotMonad.html#897" class="Function">idC</a> <a id="5258" href="Categories.Category.Cocartesian.html#2133" class="Function Operator">+₁</a> <a id="5261" href="Monad.ElgotMonad.html#4549" class="Bound">h</a><a id="5262" class="Symbol">)</a> <a id="5264" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="5266" href="Categories.Object.Coproduct.html#532" class="Function Operator">[</a> <a id="5268" href="Monad.ElgotMonad.html#7809" class="Function">T₁</a> <a id="5271" href="Categories.Object.Coproduct.html#492" class="Function">i₁</a> <a id="5274" href="Categories.Object.Coproduct.html#532" class="Function Operator">,</a> <a id="5276" href="Categories.NaturalTransformation.Core.html#783" class="Function">η.η</a> <a id="5280" class="Symbol">_</a> <a id="5282" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="5284" href="Categories.Object.Coproduct.html#512" class="Function">i₂</a> <a id="5287" href="Categories.Object.Coproduct.html#532" class="Function Operator">]</a> <a id="5289" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="5291" href="Monad.ElgotMonad.html#4541" class="Bound">f</a> <a id="5293" href="Relation.Binary.Reasoning.Base.Single.html#2564" class="Function Operator">∎</a><a id="5294" class="Symbol">))</a>
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          <a id="5356" class="Symbol">(</a><a id="5357" href="Categories.Object.Coproduct.html#532" class="Function Operator">[</a> <a id="5359" href="Monad.ElgotMonad.html#7809" class="Function">T₁</a> <a id="5362" href="Categories.Object.Coproduct.html#492" class="Function">i₁</a> <a id="5365" href="Categories.Object.Coproduct.html#532" class="Function Operator">,</a> <a id="5367" href="Categories.NaturalTransformation.Core.html#783" class="Function">η.η</a> <a id="5371" class="Symbol">_</a> <a id="5373" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="5375" href="Categories.Object.Coproduct.html#512" class="Function">i₂</a> <a id="5378" href="Categories.Object.Coproduct.html#532" class="Function Operator">]</a> <a id="5380" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="5382" class="Symbol">(((</a><a id="5385" href="Categories.Object.Coproduct.html#532" class="Function Operator">[</a> <a id="5387" href="Monad.ElgotMonad.html#7809" class="Function">T₁</a> <a id="5390" href="Categories.Object.Coproduct.html#492" class="Function">i₁</a> <a id="5393" href="Categories.Object.Coproduct.html#532" class="Function Operator">,</a> <a id="5395" href="Categories.NaturalTransformation.Core.html#783" class="Function">η.η</a> <a id="5399" class="Symbol">_</a> <a id="5401" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="5403" href="Categories.Object.Coproduct.html#512" class="Function">i₂</a> <a id="5406" href="Categories.Object.Coproduct.html#532" class="Function Operator">]</a> <a id="5408" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="5410" href="Monad.ElgotMonad.html#5330" class="Bound">f</a><a id="5411" class="Symbol">)</a> <a id="5413" href="Monad.ElgotMonad.html#2057" class="Function Operator">†</a><a id="5414" class="Symbol">)</a> <a id="5416" href="Categories.Category.Cocartesian.html#2133" class="Function Operator">+₁</a> <a id="5419" href="Monad.ElgotMonad.html#5334" class="Bound">h</a><a id="5420" class="Symbol">))</a><a id="5422" href="Monad.ElgotMonad.html#2057" class="Function Operator">†</a> <a id="5424" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">≈⟨</a> <a id="5427" href="Monad.ElgotMonad.html#2104" class="Function">†-resp-≈</a> <a id="5436" href="Categories.Category.Cocartesian.html#2662" class="Function">[]∘+₁</a> <a id="5442" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">⟩</a> 
          <a id="5455" href="Categories.Object.Coproduct.html#532" class="UnsolvedMeta Function Operator">[</a><a id="5456" class="UnsolvedMeta"> </a><a id="5457" href="Monad.ElgotMonad.html#7809" class="UnsolvedMeta Function">T₁</a><a id="5459" class="UnsolvedMeta"> </a><a id="5460" href="Categories.Object.Coproduct.html#492" class="UnsolvedMeta Function">i₁</a><a id="5462" class="UnsolvedMeta"> </a><a id="5463" href="Categories.Category.Core.html#656" class="UnsolvedMeta Function Operator">∘</a><a id="5464" class="UnsolvedMeta"> </a><a id="5465" class="UnsolvedMeta Symbol">(</a><a id="5466" href="Categories.Object.Coproduct.html#532" class="UnsolvedMeta Function Operator">[</a><a id="5467" class="UnsolvedMeta"> </a><a id="5468" href="Monad.ElgotMonad.html#7809" class="UnsolvedMeta Function">T₁</a><a id="5470" class="UnsolvedMeta"> </a><a id="5471" href="Categories.Object.Coproduct.html#492" class="UnsolvedMeta Function">i₁</a><a id="5473" class="UnsolvedMeta"> </a><a id="5474" href="Categories.Object.Coproduct.html#532" class="UnsolvedMeta Function Operator">,</a><a id="5475" class="UnsolvedMeta"> </a><a id="5476" href="Categories.NaturalTransformation.Core.html#783" class="UnsolvedMeta Function">η.η</a><a id="5479" class="UnsolvedMeta"> </a><a id="5480" class="UnsolvedMeta Symbol">_</a><a id="5481" class="UnsolvedMeta"> </a><a id="5482" href="Categories.Category.Core.html#656" class="UnsolvedMeta Function Operator">∘</a><a id="5483" class="UnsolvedMeta"> </a><a id="5484" href="Categories.Object.Coproduct.html#512" class="UnsolvedMeta Function">i₂</a><a id="5486" class="UnsolvedMeta"> </a><a id="5487" href="Categories.Object.Coproduct.html#532" class="UnsolvedMeta Function Operator">]</a><a id="5488" class="UnsolvedMeta"> </a><a id="5489" href="Categories.Category.Core.html#656" class="UnsolvedMeta Function Operator">∘</a><a id="5490" class="UnsolvedMeta"> </a><a id="5491" href="Monad.ElgotMonad.html#5330" class="UnsolvedMeta Bound">f</a><a id="5492" class="UnsolvedMeta Symbol">)</a><a id="5493" class="UnsolvedMeta"> </a><a id="5494" href="Monad.ElgotMonad.html#2057" class="UnsolvedMeta Function Operator">†</a><a id="5495" class="UnsolvedMeta"> </a><a id="5496" href="Categories.Object.Coproduct.html#532" class="UnsolvedMeta Function Operator">,</a><a id="5497" class="UnsolvedMeta"> </a><a id="5498" class="UnsolvedMeta Symbol">(</a><a id="5499" href="Categories.NaturalTransformation.Core.html#783" class="UnsolvedMeta Function">η.η</a><a id="5502" class="UnsolvedMeta"> </a><a id="5503" class="UnsolvedMeta Symbol">_</a><a id="5504" class="UnsolvedMeta"> </a><a id="5505" href="Categories.Category.Core.html#656" class="UnsolvedMeta Function Operator">∘</a><a id="5506" class="UnsolvedMeta"> </a><a id="5507" href="Categories.Object.Coproduct.html#512" class="UnsolvedMeta Function">i₂</a><a id="5509" class="UnsolvedMeta Symbol">)</a><a id="5510" class="UnsolvedMeta"> </a><a id="5511" href="Categories.Category.Core.html#656" class="UnsolvedMeta Function Operator">∘</a><a id="5512" class="UnsolvedMeta"> </a><a id="5513" href="Monad.ElgotMonad.html#5334" class="UnsolvedMeta Bound">h</a><a id="5514" class="UnsolvedMeta"> </a><a id="5515" href="Categories.Object.Coproduct.html#532" class="UnsolvedMeta Function Operator">]</a><a id="5516" class="UnsolvedMeta"> </a><a id="5517" href="Monad.ElgotMonad.html#2057" class="UnsolvedMeta Function Operator">†</a><a id="5518" class="UnsolvedMeta"> </a><a id="5519" href="Relation.Binary.Reasoning.Setoid.html#1061" class="UnsolvedMeta Function">≈⟨</a><a id="5521" class="UnsolvedMeta"> </a><a id="5522" class="UnsolvedMeta Hole">{!   !}</a><a id="5529" class="UnsolvedMeta"> </a><a id="5530" href="Relation.Binary.Reasoning.Setoid.html#1061" class="UnsolvedMeta Function">⟩</a>
          <a id="5542" class="Hole">{!   !}</a> <a id="5550" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">≈⟨</a> <a id="5553" class="Hole">{!   !}</a> <a id="5561" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">⟩</a>
          <a id="5573" href="Categories.Object.Coproduct.html#532" class="Function Operator">[</a> <a id="5575" href="Categories.Object.Coproduct.html#532" class="Function Operator">[</a> <a id="5577" href="Monad.ElgotMonad.html#7809" class="Function">T₁</a> <a id="5580" href="Categories.Object.Coproduct.html#492" class="Function">i₁</a> <a id="5583" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="5585" href="Monad.ElgotMonad.html#897" class="Function">idC</a> <a id="5589" href="Categories.Object.Coproduct.html#532" class="Function Operator">,</a> <a id="5591" class="Symbol">(</a><a id="5592" href="Categories.NaturalTransformation.Core.html#783" class="Function">η.η</a> <a id="5596" class="Symbol">_</a> <a id="5598" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="5600" href="Categories.Object.Coproduct.html#512" class="Function">i₂</a><a id="5602" class="Symbol">)</a> <a id="5604" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="5606" href="Categories.Object.Coproduct.html#492" class="Function">i₁</a> <a id="5609" href="Categories.Object.Coproduct.html#532" class="Function Operator">]</a> <a id="5611" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="5613" href="Monad.ElgotMonad.html#5330" class="Bound">f</a> <a id="5615" href="Categories.Object.Coproduct.html#532" class="Function Operator">,</a> <a id="5617" class="Symbol">(</a><a id="5618" href="Categories.NaturalTransformation.Core.html#783" class="Function">η.η</a> <a id="5622" class="Symbol">_</a> <a id="5624" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="5626" href="Categories.Object.Coproduct.html#512" class="Function">i₂</a><a id="5628" class="Symbol">)</a> <a id="5630" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="5632" href="Monad.ElgotMonad.html#5334" class="Bound">h</a> <a id="5634" href="Categories.Object.Coproduct.html#532" class="Function Operator">]</a> <a id="5636" href="Monad.ElgotMonad.html#2057" class="Function Operator">†</a> <a id="5638" href="Relation.Binary.Reasoning.Setoid.html#1162" class="Function">≈˘⟨</a> <a id="5642" href="Monad.ElgotMonad.html#2104" class="Function">†-resp-≈</a> <a id="5651" class="Symbol">(</a><a id="5652" href="Categories.Object.Coproduct.html#816" class="Function">[]-cong₂</a> <a id="5661" class="Symbol">(</a><a id="5662" href="Categories.Morphism.Reasoning.Core.html#2048" class="Function">pullˡ</a> <a id="5668" href="Categories.Category.Cocartesian.html#2662" class="Function">[]∘+₁</a><a id="5673" class="Symbol">)</a> <a id="5675" class="Symbol">(</a><a id="5676" href="Categories.Morphism.Reasoning.Core.html#2048" class="Function">pullˡ</a> <a id="5682" href="Categories.Object.Coproduct.html#602" class="Function">inject₂</a><a id="5689" class="Symbol">))</a> <a id="5692" href="Relation.Binary.Reasoning.Setoid.html#1162" class="Function">⟩</a>
          <a id="5704" href="Categories.Object.Coproduct.html#532" class="Function Operator">[</a> <a id="5706" href="Categories.Object.Coproduct.html#532" class="Function Operator">[</a> <a id="5708" href="Monad.ElgotMonad.html#7809" class="Function">T₁</a> <a id="5711" href="Categories.Object.Coproduct.html#492" class="Function">i₁</a> <a id="5714" href="Categories.Object.Coproduct.html#532" class="Function Operator">,</a> <a id="5716" href="Categories.NaturalTransformation.Core.html#783" class="Function">η.η</a> <a id="5720" class="Symbol">_</a> <a id="5722" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="5724" href="Categories.Object.Coproduct.html#512" class="Function">i₂</a> <a id="5727" href="Categories.Object.Coproduct.html#532" class="Function Operator">]</a> <a id="5729" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="5731" class="Symbol">(</a><a id="5732" href="Monad.ElgotMonad.html#897" class="Function">idC</a> <a id="5736" href="Categories.Category.Cocartesian.html#2133" class="Function Operator">+₁</a> <a id="5739" href="Categories.Object.Coproduct.html#492" class="Function">i₁</a><a id="5741" class="Symbol">)</a> <a id="5743" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="5745" href="Monad.ElgotMonad.html#5330" class="Bound">f</a> <a id="5747" href="Categories.Object.Coproduct.html#532" class="Function Operator">,</a> <a id="5749" href="Categories.Object.Coproduct.html#532" class="Function Operator">[</a> <a id="5751" href="Monad.ElgotMonad.html#7809" class="Function">T₁</a> <a id="5754" href="Categories.Object.Coproduct.html#492" class="Function">i₁</a> <a id="5757" href="Categories.Object.Coproduct.html#532" class="Function Operator">,</a> <a id="5759" href="Categories.NaturalTransformation.Core.html#783" class="Function">η.η</a> <a id="5763" class="Symbol">_</a> <a id="5765" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="5767" href="Categories.Object.Coproduct.html#512" class="Function">i₂</a> <a id="5770" href="Categories.Object.Coproduct.html#532" class="Function Operator">]</a> <a id="5772" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="5774" href="Categories.Object.Coproduct.html#512" class="Function">i₂</a> <a id="5777" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="5779" href="Monad.ElgotMonad.html#5334" class="Bound">h</a> <a id="5781" href="Categories.Object.Coproduct.html#532" class="Function Operator">]</a> <a id="5783" href="Monad.ElgotMonad.html#2057" class="Function Operator">†</a> <a id="5785" href="Relation.Binary.Reasoning.Setoid.html#1162" class="Function">≈˘⟨</a> <a id="5789" href="Monad.ElgotMonad.html#2104" class="Function">†-resp-≈</a> <a id="5798" href="Categories.Category.Cocartesian.html#2736" class="Function">∘[]</a> <a id="5802" href="Relation.Binary.Reasoning.Setoid.html#1162" class="Function">⟩</a>
          <a id="5814" class="Symbol">(</a><a id="5815" href="Categories.Object.Coproduct.html#532" class="Function Operator">[</a> <a id="5817" href="Monad.ElgotMonad.html#7809" class="Function">T₁</a> <a id="5820" href="Categories.Object.Coproduct.html#492" class="Function">i₁</a> <a id="5823" href="Categories.Object.Coproduct.html#532" class="Function Operator">,</a> <a id="5825" href="Categories.NaturalTransformation.Core.html#783" class="Function">η.η</a> <a id="5829" class="Symbol">_</a> <a id="5831" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="5833" href="Categories.Object.Coproduct.html#512" class="Function">i₂</a> <a id="5836" href="Categories.Object.Coproduct.html#532" class="Function Operator">]</a> <a id="5838" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="5840" href="Categories.Object.Coproduct.html#532" class="Function Operator">[</a> <a id="5842" class="Symbol">(</a><a id="5843" href="Monad.ElgotMonad.html#897" class="Function">idC</a> <a id="5847" href="Categories.Category.Cocartesian.html#2133" class="Function Operator">+₁</a> <a id="5850" href="Categories.Object.Coproduct.html#492" class="Function">i₁</a><a id="5852" class="Symbol">)</a> <a id="5854" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="5856" href="Monad.ElgotMonad.html#5330" class="Bound">f</a> <a id="5858" href="Categories.Object.Coproduct.html#532" class="Function Operator">,</a> <a id="5860" href="Categories.Object.Coproduct.html#512" class="Function">i₂</a> <a id="5863" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="5865" href="Monad.ElgotMonad.html#5334" class="Bound">h</a> <a id="5867" href="Categories.Object.Coproduct.html#532" class="Function Operator">]</a><a id="5868" class="Symbol">)</a><a id="5869" href="Monad.ElgotMonad.html#2057" class="Function Operator">†</a> <a id="5871" href="Relation.Binary.Reasoning.Base.Single.html#2564" class="Function Operator">∎</a>
        <a id="5881" class="Symbol">;</a> <a id="5883" href="ElgotAlgebra.html#2582" class="Field">#-resp-≈</a> <a id="5892" class="Symbol">=</a> <a id="5894" class="Symbol">λ</a> <a id="5896" href="Monad.ElgotMonad.html#5896" class="Bound">fg</a> <a id="5899" class="Symbol">→</a> <a id="5901" href="Monad.ElgotMonad.html#2104" class="Function">†-resp-≈</a> <a id="5910" class="Symbol">(</a><a id="5911" href="Categories.Category.Core.html#1706" class="Function">∘-resp-≈ʳ</a> <a id="5921" href="Monad.ElgotMonad.html#5896" class="Bound">fg</a><a id="5923" class="Symbol">)</a>
        <a id="5933" class="Symbol">}</a>
      <a id="5941" class="Symbol">;</a> <a id="5943" href="Monad.ElgotMonad.html#1543" class="Field">assoc</a> <a id="5949" class="Symbol">=</a> <a id="5951" class="Symbol">λ</a> <a id="5953" class="Symbol">{</a><a id="5954" href="Monad.ElgotMonad.html#5954" class="Bound">X</a><a id="5955" class="Symbol">}</a> <a id="5957" class="Symbol">{</a><a id="5958" href="Monad.ElgotMonad.html#5958" class="Bound">Y</a><a id="5959" class="Symbol">}</a> <a id="5961" class="Symbol">{</a><a id="5962" href="Monad.ElgotMonad.html#5962" class="Bound">Z</a><a id="5963" class="Symbol">}</a> <a id="5965" href="Monad.ElgotMonad.html#5965" class="Bound">f</a> <a id="5967" href="Monad.ElgotMonad.html#5967" class="Bound">h</a> <a id="5969" class="Symbol">→</a> <a id="5971" href="Relation.Binary.Reasoning.Base.Single.html#1925" class="Function Operator">begin</a> 
        <a id="5986" class="Comment">-- TODO tidy up by moving doing sym outside, apply Naturality and then do `†-resp-≈ pullˡ` once.</a>
        <a id="6091" class="Symbol">(</a><a id="6092" href="Categories.Object.Coproduct.html#532" class="Function Operator">[</a> <a id="6094" href="Monad.ElgotMonad.html#7809" class="Function">T₁</a> <a id="6097" href="Categories.Object.Coproduct.html#492" class="Function">i₁</a> <a id="6100" href="Categories.Object.Coproduct.html#532" class="Function Operator">,</a> <a id="6102" href="Categories.NaturalTransformation.Core.html#783" class="Function">η.η</a> <a id="6106" class="Symbol">_</a> <a id="6108" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="6110" href="Categories.Object.Coproduct.html#512" class="Function">i₂</a> <a id="6113" href="Categories.Object.Coproduct.html#532" class="Function Operator">]</a> <a id="6115" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="6117" class="Symbol">(</a><a id="6118" href="Categories.NaturalTransformation.Core.html#783" class="Function">μ.η</a> <a id="6122" href="Monad.ElgotMonad.html#5958" class="Bound">Y</a> <a id="6124" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="6126" href="Monad.ElgotMonad.html#7809" class="Function">T₁</a> <a id="6129" href="Monad.ElgotMonad.html#5967" class="Bound">h</a> <a id="6131" href="Categories.Category.Cocartesian.html#2133" class="Function Operator">+₁</a> <a id="6134" href="Monad.ElgotMonad.html#897" class="Function">idC</a><a id="6137" class="Symbol">)</a> <a id="6139" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="6141" href="Monad.ElgotMonad.html#5965" class="Bound">f</a><a id="6142" class="Symbol">)</a><a id="6143" href="Monad.ElgotMonad.html#2057" class="Function Operator">†</a>                                                             <a id="6205" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">≈⟨</a> <a id="6208" href="Monad.ElgotMonad.html#2104" class="Function">†-resp-≈</a> <a id="6217" class="Symbol">(</a><a id="6218" href="Categories.Morphism.Reasoning.Core.html#2048" class="Function">pullˡ</a> <a id="6224" href="Categories.Category.Cocartesian.html#2662" class="Function">[]∘+₁</a><a id="6229" class="Symbol">)</a> <a id="6231" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">⟩</a> 
        <a id="6242" class="Symbol">((</a><a id="6244" href="Categories.Object.Coproduct.html#532" class="Function Operator">[</a> <a id="6246" href="Monad.ElgotMonad.html#7809" class="Function">T₁</a> <a id="6249" href="Categories.Object.Coproduct.html#492" class="Function">i₁</a> <a id="6252" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="6254" href="Categories.NaturalTransformation.Core.html#783" class="Function">μ.η</a> <a id="6258" class="Symbol">_</a> <a id="6260" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="6262" href="Monad.ElgotMonad.html#7809" class="Function">T₁</a> <a id="6265" href="Monad.ElgotMonad.html#5967" class="Bound">h</a> <a id="6267" href="Categories.Object.Coproduct.html#532" class="Function Operator">,</a> <a id="6269" class="Symbol">(</a><a id="6270" href="Categories.NaturalTransformation.Core.html#783" class="Function">η.η</a> <a id="6274" class="Symbol">_</a> <a id="6276" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="6278" href="Categories.Object.Coproduct.html#512" class="Function">i₂</a><a id="6280" class="Symbol">)</a> <a id="6282" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="6284" href="Monad.ElgotMonad.html#897" class="Function">idC</a> <a id="6288" href="Categories.Object.Coproduct.html#532" class="Function Operator">]</a> <a id="6290" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="6292" href="Monad.ElgotMonad.html#5965" class="Bound">f</a><a id="6293" class="Symbol">)</a><a id="6294" href="Monad.ElgotMonad.html#2057" class="Function Operator">†</a><a id="6295" class="Symbol">)</a>                                                            <a id="6356" href="Relation.Binary.Reasoning.Setoid.html#1162" class="Function">≈˘⟨</a> <a id="6360" href="Monad.ElgotMonad.html#2104" class="Function">†-resp-≈</a> <a id="6369" class="Symbol">(</a><a id="6370" href="Categories.Category.Core.html#1597" class="Function">∘-resp-≈ˡ</a> <a id="6380" class="Symbol">(</a><a id="6381" href="Categories.Object.Coproduct.html#816" class="Function">[]-cong₂</a> <a id="6390" href="Categories.Category.Core.html#715" class="Function">assoc</a> <a id="6396" class="Symbol">(</a><a id="6397" href="Relation.Binary.Structures.html#1603" class="Function">sym</a> <a id="6401" href="Categories.Category.Core.html#1145" class="Function">identityʳ</a><a id="6410" class="Symbol">)))</a> <a id="6414" href="Relation.Binary.Reasoning.Setoid.html#1162" class="Function">⟩</a>
        <a id="6424" class="Symbol">(</a><a id="6425" href="Categories.Object.Coproduct.html#532" class="Function Operator">[</a> <a id="6427" class="Symbol">(</a><a id="6428" href="Monad.ElgotMonad.html#7809" class="Function">T₁</a> <a id="6431" href="Categories.Object.Coproduct.html#492" class="Function">i₁</a> <a id="6434" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="6436" href="Categories.NaturalTransformation.Core.html#783" class="Function">μ.η</a> <a id="6440" class="Symbol">_)</a> <a id="6443" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="6445" href="Monad.ElgotMonad.html#7809" class="Function">T₁</a> <a id="6448" href="Monad.ElgotMonad.html#5967" class="Bound">h</a> <a id="6450" href="Categories.Object.Coproduct.html#532" class="Function Operator">,</a> <a id="6452" href="Categories.NaturalTransformation.Core.html#783" class="Function">η.η</a> <a id="6456" class="Symbol">_</a> <a id="6458" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="6460" href="Categories.Object.Coproduct.html#512" class="Function">i₂</a> <a id="6463" href="Categories.Object.Coproduct.html#532" class="Function Operator">]</a> <a id="6465" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="6467" href="Monad.ElgotMonad.html#5965" class="Bound">f</a><a id="6468" class="Symbol">)</a><a id="6469" href="Monad.ElgotMonad.html#2057" class="Function Operator">†</a>                                                                    <a id="6538" href="Relation.Binary.Reasoning.Setoid.html#1162" class="Function">≈˘⟨</a> <a id="6542" href="Monad.ElgotMonad.html#2104" class="Function">†-resp-≈</a> <a id="6551" class="Symbol">(</a><a id="6552" href="Categories.Category.Core.html#1597" class="Function">∘-resp-≈ˡ</a> <a id="6562" class="Symbol">(</a><a id="6563" href="Categories.Category.Cocartesian.html#2271" class="Function">[]-congʳ</a> <a id="6572" class="Symbol">(</a><a id="6573" href="Categories.Morphism.Reasoning.Core.html#2048" class="Function">pullˡ</a> <a id="6579" class="Symbol">(</a><a id="6580" href="Categories.NaturalTransformation.Core.html#827" class="Function">μ.commute</a> <a id="6590" class="Symbol">_))))</a> <a id="6596" href="Relation.Binary.Reasoning.Setoid.html#1162" class="Function">⟩</a>
        <a id="6606" class="Symbol">(</a><a id="6607" href="Categories.Object.Coproduct.html#532" class="Function Operator">[</a> <a id="6609" href="Categories.NaturalTransformation.Core.html#783" class="Function">μ.η</a> <a id="6613" class="Symbol">_</a> <a id="6615" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="6617" href="Monad.ElgotMonad.html#7809" class="Function">T₁</a> <a id="6620" class="Symbol">(</a><a id="6621" href="Monad.ElgotMonad.html#7809" class="Function">T₁</a> <a id="6624" href="Categories.Object.Coproduct.html#492" class="Function">i₁</a><a id="6626" class="Symbol">)</a> <a id="6628" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="6630" href="Monad.ElgotMonad.html#7809" class="Function">T₁</a> <a id="6633" href="Monad.ElgotMonad.html#5967" class="Bound">h</a> <a id="6635" href="Categories.Object.Coproduct.html#532" class="Function Operator">,</a> <a id="6637" href="Categories.NaturalTransformation.Core.html#783" class="Function">η.η</a> <a id="6641" class="Symbol">_</a> <a id="6643" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="6645" href="Categories.Object.Coproduct.html#512" class="Function">i₂</a> <a id="6648" href="Categories.Object.Coproduct.html#532" class="Function Operator">]</a> <a id="6650" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="6652" href="Monad.ElgotMonad.html#5965" class="Bound">f</a><a id="6653" class="Symbol">)</a><a id="6654" href="Monad.ElgotMonad.html#2057" class="Function Operator">†</a>                                                                 <a id="6720" href="Relation.Binary.Reasoning.Setoid.html#1162" class="Function">≈˘⟨</a> <a id="6724" href="Monad.ElgotMonad.html#2104" class="Function">†-resp-≈</a> <a id="6733" class="Symbol">(</a><a id="6734" href="Categories.Category.Core.html#1597" class="Function">∘-resp-≈ˡ</a> <a id="6744" class="Symbol">(</a><a id="6745" href="Categories.Object.Coproduct.html#816" class="Function">[]-cong₂</a> <a id="6754" class="Symbol">(</a><a id="6755" href="Categories.Category.Core.html#1706" class="Function">∘-resp-≈ʳ</a> <a id="6765" href="Categories.Functor.Core.html#565" class="Function">homomorphism</a><a id="6777" class="Symbol">)</a> <a id="6779" class="Symbol">(</a><a id="6780" href="Categories.Morphism.Reasoning.Core.html#6851" class="Function">cancelˡ</a> <a id="6788" href="Categories.Monad.html#953" class="Function">T.identityʳ</a><a id="6799" class="Symbol">)))</a> <a id="6803" href="Relation.Binary.Reasoning.Setoid.html#1162" class="Function">⟩</a>
        <a id="6813" class="Symbol">(</a><a id="6814" href="Categories.Object.Coproduct.html#532" class="Function Operator">[</a> <a id="6816" href="Categories.NaturalTransformation.Core.html#783" class="Function">μ.η</a> <a id="6820" class="Symbol">_</a> <a id="6822" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="6824" href="Monad.ElgotMonad.html#7809" class="Function">T₁</a> <a id="6827" class="Symbol">(</a><a id="6828" href="Monad.ElgotMonad.html#7809" class="Function">T₁</a> <a id="6831" href="Categories.Object.Coproduct.html#492" class="Function">i₁</a> <a id="6834" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="6836" href="Monad.ElgotMonad.html#5967" class="Bound">h</a><a id="6837" class="Symbol">)</a> <a id="6839" href="Categories.Object.Coproduct.html#532" class="Function Operator">,</a> <a id="6841" href="Categories.NaturalTransformation.Core.html#783" class="Function">μ.η</a> <a id="6845" class="Symbol">_</a> <a id="6847" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="6849" href="Categories.NaturalTransformation.Core.html#783" class="Function">η.η</a> <a id="6853" class="Symbol">_</a> <a id="6855" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="6857" href="Categories.NaturalTransformation.Core.html#783" class="Function">η.η</a> <a id="6861" class="Symbol">_</a> <a id="6863" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="6865" href="Categories.Object.Coproduct.html#512" class="Function">i₂</a> <a id="6868" href="Categories.Object.Coproduct.html#532" class="Function Operator">]</a> <a id="6870" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="6872" href="Monad.ElgotMonad.html#5965" class="Bound">f</a><a id="6873" class="Symbol">)</a><a id="6874" href="Monad.ElgotMonad.html#2057" class="Function Operator">†</a>                                                    <a id="6927" href="Relation.Binary.Reasoning.Setoid.html#1162" class="Function">≈˘⟨</a> <a id="6931" href="Monad.ElgotMonad.html#2104" class="Function">†-resp-≈</a> <a id="6940" class="Symbol">(</a><a id="6941" href="Categories.Category.Core.html#1597" class="Function">∘-resp-≈ˡ</a> <a id="6951" class="Symbol">(</a><a id="6952" href="Categories.Object.Coproduct.html#816" class="Function">[]-cong₂</a> <a id="6961" class="Symbol">(</a><a id="6962" href="Categories.Category.Core.html#1706" class="Function">∘-resp-≈ʳ</a> <a id="6972" class="Symbol">(</a><a id="6973" href="Categories.Functor.Core.html#696" class="Function">F-resp-≈</a> <a id="6982" href="Categories.Object.Coproduct.html#569" class="Function">inject₁</a><a id="6989" class="Symbol">))</a> <a id="6992" class="Symbol">(</a><a id="6993" href="Categories.Category.Core.html#1706" class="Function">∘-resp-≈ʳ</a> <a id="7003" class="Symbol">(</a><a id="7004" href="Categories.Morphism.Reasoning.Core.html#1914" class="Function">pullʳ</a> <a id="7010" href="Categories.Object.Coproduct.html#602" class="Function">inject₂</a><a id="7017" class="Symbol">))))</a> <a id="7022" href="Relation.Binary.Reasoning.Setoid.html#1162" class="Function">⟩</a>
        <a id="7032" class="Symbol">(</a><a id="7033" href="Categories.Object.Coproduct.html#532" class="Function Operator">[</a> <a id="7035" href="Categories.NaturalTransformation.Core.html#783" class="Function">μ.η</a> <a id="7039" class="Symbol">_</a> <a id="7041" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="7043" href="Monad.ElgotMonad.html#7809" class="Function">T₁</a> <a id="7046" class="Symbol">(</a><a id="7047" href="Categories.Object.Coproduct.html#532" class="Function Operator">[</a> <a id="7049" href="Monad.ElgotMonad.html#7809" class="Function">T₁</a> <a id="7052" href="Categories.Object.Coproduct.html#492" class="Function">i₁</a> <a id="7055" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="7057" href="Monad.ElgotMonad.html#5967" class="Bound">h</a> <a id="7059" href="Categories.Object.Coproduct.html#532" class="Function Operator">,</a> <a id="7061" href="Categories.NaturalTransformation.Core.html#783" class="Function">η.η</a> <a id="7065" class="Symbol">_</a> <a id="7067" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="7069" href="Categories.Object.Coproduct.html#512" class="Function">i₂</a> <a id="7072" href="Categories.Object.Coproduct.html#532" class="Function Operator">]</a> <a id="7074" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="7076" href="Categories.Object.Coproduct.html#492" class="Function">i₁</a><a id="7078" class="Symbol">)</a> <a id="7080" href="Categories.Object.Coproduct.html#532" class="Function Operator">,</a> <a id="7082" href="Categories.NaturalTransformation.Core.html#783" class="Function">μ.η</a> <a id="7086" class="Symbol">_</a> <a id="7088" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="7090" class="Symbol">(</a><a id="7091" href="Categories.NaturalTransformation.Core.html#783" class="Function">η.η</a> <a id="7095" class="Symbol">_</a> <a id="7097" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="7099" href="Categories.Object.Coproduct.html#532" class="Function Operator">[</a> <a id="7101" href="Monad.ElgotMonad.html#7809" class="Function">T₁</a> <a id="7104" href="Categories.Object.Coproduct.html#492" class="Function">i₁</a> <a id="7107" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="7109" href="Monad.ElgotMonad.html#5967" class="Bound">h</a> <a id="7111" href="Categories.Object.Coproduct.html#532" class="Function Operator">,</a> <a id="7113" href="Categories.NaturalTransformation.Core.html#783" class="Function">η.η</a> <a id="7117" class="Symbol">_</a> <a id="7119" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="7121" href="Categories.Object.Coproduct.html#512" class="Function">i₂</a> <a id="7124" href="Categories.Object.Coproduct.html#532" class="Function Operator">]</a><a id="7125" class="Symbol">)</a> <a id="7127" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="7129" href="Categories.Object.Coproduct.html#512" class="Function">i₂</a> <a id="7132" href="Categories.Object.Coproduct.html#532" class="Function Operator">]</a> <a id="7134" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="7136" href="Monad.ElgotMonad.html#5965" class="Bound">f</a><a id="7137" class="Symbol">)</a><a id="7138" href="Monad.ElgotMonad.html#2057" class="Function Operator">†</a>       <a id="7146" href="Relation.Binary.Reasoning.Setoid.html#1162" class="Function">≈˘⟨</a> <a id="7150" href="Monad.ElgotMonad.html#2104" class="Function">†-resp-≈</a> <a id="7159" class="Symbol">(</a><a id="7160" href="Categories.Category.Core.html#1597" class="Function">∘-resp-≈ˡ</a> <a id="7170" class="Symbol">(</a><a id="7171" href="Categories.Object.Coproduct.html#816" class="Function">[]-cong₂</a> <a id="7180" class="Symbol">(</a><a id="7181" href="Categories.Morphism.Reasoning.Core.html#1914" class="Function">pullʳ</a> <a id="7187" class="Symbol">(</a><a id="7188" href="Relation.Binary.Structures.html#1603" class="Function">sym</a> <a id="7192" href="Categories.Functor.Core.html#565" class="Function">homomorphism</a><a id="7204" class="Symbol">))</a> <a id="7207" class="Symbol">(</a><a id="7208" href="Categories.Morphism.Reasoning.Core.html#1914" class="Function">pullʳ</a> <a id="7214" class="Symbol">(</a><a id="7215" href="Categories.Morphism.Reasoning.Core.html#2048" class="Function">pullˡ</a> <a id="7221" class="Symbol">(</a><a id="7222" href="Categories.NaturalTransformation.Core.html#1043" class="Function">η.sym-commute</a> <a id="7236" class="Symbol">_)))))</a> <a id="7243" href="Relation.Binary.Reasoning.Setoid.html#1162" class="Function">⟩</a>
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        <a id="7541" class="Symbol">(</a><a id="7542" href="Categories.NaturalTransformation.Core.html#783" class="Function">μ.η</a> <a id="7546" href="Monad.ElgotMonad.html#5958" class="Bound">Y</a> <a id="7548" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="7550" href="Monad.ElgotMonad.html#7809" class="Function">T₁</a> <a id="7553" href="Monad.ElgotMonad.html#5967" class="Bound">h</a><a id="7554" class="Symbol">)</a> <a id="7556" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="7558" class="Symbol">(</a><a id="7559" href="Categories.Object.Coproduct.html#532" class="Function Operator">[</a> <a id="7561" href="Monad.ElgotMonad.html#7809" class="Function">T₁</a> <a id="7564" href="Categories.Object.Coproduct.html#492" class="Function">i₁</a> <a id="7567" href="Categories.Object.Coproduct.html#532" class="Function Operator">,</a>  <a id="7570" href="Categories.NaturalTransformation.Core.html#783" class="Function">η.η</a> <a id="7574" class="Symbol">_</a> <a id="7576" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="7578" href="Categories.Object.Coproduct.html#512" class="Function">i₂</a> <a id="7581" href="Categories.Object.Coproduct.html#532" class="Function Operator">]</a> <a id="7583" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="7585" href="Monad.ElgotMonad.html#5965" class="Bound">f</a><a id="7586" class="Symbol">)</a><a id="7587" href="Monad.ElgotMonad.html#2057" class="Function Operator">†</a>                                                                   <a id="7655" href="Relation.Binary.Reasoning.Base.Single.html#2564" class="Function Operator">∎</a> 
      <a id="7664" class="Symbol">}</a> 
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    <a id="7677" class="Keyword">where</a>
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</pre>