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<pre class="Agda"><a id="649" class="Keyword">module</a> <a id="656" href="Monad.Instance.Delay.html" class="Module">Monad.Instance.Delay</a> <a id="677" class="Symbol">{</a><a id="678" href="Monad.Instance.Delay.html#678" class="Bound">o</a> <a id="680" href="Monad.Instance.Delay.html#680" class="Bound">ℓ</a> <a id="682" href="Monad.Instance.Delay.html#682" class="Bound">e</a><a id="683" class="Symbol">}</a> <a id="685" class="Symbol">(</a><a id="686" href="Monad.Instance.Delay.html#686" class="Bound">ambient</a> <a id="694" class="Symbol">:</a> <a id="696" href="Category.Ambient.html#1681" class="Record">Ambient</a> <a id="704" href="Monad.Instance.Delay.html#678" class="Bound">o</a> <a id="706" href="Monad.Instance.Delay.html#680" class="Bound">ℓ</a> <a id="708" href="Monad.Instance.Delay.html#682" class="Bound">e</a><a id="709" class="Symbol">)</a> <a id="711" class="Keyword">where</a>
<a id="719" class="Keyword">open</a> <a id="724" href="Category.Ambient.html#1681" class="Module">Ambient</a> <a id="732" href="Monad.Instance.Delay.html#686" class="Bound">ambient</a>
</pre>
<h2 id="definition">Definition</h2>
<p>The delay monad is usually defined as a coinductive type with two
constructors <code>now : X → D X</code> and
<code>later : D X → D X</code>, e.g. in the <a
href="https://agda.github.io/agda-stdlib/Effect.Monad.Partiality.html#1523">agda-stdlib</a></p>
<p>We will now define it categorically by existence of final coalgebras
for the functor <code>(X + -)</code> where <code>X</code> is some
object. This functor trivially sends objects <code>Y</code> to
<code>X + Y</code> and functions <code>f</code> to
<code>id + f</code>.</p>
<p>In this definition <code>D X</code> is the underlying object of the
final coalgebra given by <code>X</code>. We then use Lambek’s Lemma to
gain an isomorphism <code>D X ≅ X + D X</code>, whose inverse can be
factored into the constructors <code>now</code> and
<code>later</code>.</p>
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<pre class="Agda"> <a id="1685" class="Keyword">record</a> <a id="DelayM"></a><a id="1692" href="Monad.Instance.Delay.html#1692" class="Record">DelayM</a> <a id="1699" class="Symbol">:</a> <a id="1701" href="Agda.Primitive.html#388" class="Primitive">Set</a> <a id="1705" class="Symbol">(</a><a id="1706" href="Monad.Instance.Delay.html#678" class="Bound">o</a> <a id="1708" href="Agda.Primitive.html#961" class="Primitive Operator">⊔</a> <a id="1710" href="Monad.Instance.Delay.html#680" class="Bound">ℓ</a> <a id="1712" href="Agda.Primitive.html#961" class="Primitive Operator">⊔</a> <a id="1714" href="Monad.Instance.Delay.html#682" class="Bound">e</a><a id="1715" class="Symbol">)</a> <a id="1717" class="Keyword">where</a>
<a id="1727" class="Keyword">field</a>
<a id="DelayM.coalgebras"></a><a id="1739" href="Monad.Instance.Delay.html#1739" class="Field">coalgebras</a> <a id="1750" class="Symbol">:</a> <a id="1752" class="Symbol">∀</a> <a id="1754" class="Symbol">(</a><a id="1755" href="Monad.Instance.Delay.html#1755" class="Bound">A</a> <a id="1757" class="Symbol">:</a> <a id="1759" href="Categories.Category.Core.html#559" class="Function">Obj</a><a id="1762" class="Symbol">)</a> <a id="1764" class="Symbol">→</a> <a id="1766" href="Categories.Object.Terminal.html#860" class="Record">Terminal</a> <a id="1775" class="Symbol">(</a><a id="1776" href="Categories.Category.Construction.F-Coalgebras.html#598" class="Function">F-Coalgebras</a> <a id="1789" class="Symbol">(</a><a id="1790" href="Monad.Instance.Delay.html#1755" class="Bound">A</a> <a id="1792" href="Categories.Category.Cocartesian.html#3335" class="Function Operator">+-</a><a id="1794" class="Symbol">))</a>
<a id="1802" class="Keyword">module</a> <a id="1809" href="Monad.Instance.Delay.html#1809" class="Module">_</a> <a id="1811" class="Symbol">{</a><a id="1812" href="Monad.Instance.Delay.html#1812" class="Bound">X</a> <a id="1814" class="Symbol">:</a> <a id="1816" href="Categories.Category.Core.html#559" class="Function">Obj</a><a id="1819" class="Symbol">}</a> <a id="1821" class="Keyword">where</a>
<a id="1833" class="Keyword">open</a> <a id="1838" href="Categories.Object.Terminal.html#860" class="Module">Terminal</a> <a id="1847" class="Symbol">(</a><a id="1848" href="Monad.Instance.Delay.html#1739" class="Field">coalgebras</a> <a id="1859" href="Monad.Instance.Delay.html#1812" class="Bound">X</a><a id="1860" class="Symbol">)</a> <a id="1862" class="Keyword">using</a> <a id="1868" class="Symbol">(</a><a id="1869" href="Categories.Object.Terminal.html#905" class="Field">⊤</a><a id="1870" class="Symbol">;</a> <a id="1872" href="Categories.Object.Terminal.html#577" class="Function">!</a><a id="1873" class="Symbol">;</a> <a id="1875" href="Categories.Object.Terminal.html#605" class="Function">!-unique</a><a id="1883" class="Symbol">)</a>
<a id="1891" class="Keyword">open</a> <a id="1896" href="Categories.Functor.Coalgebra.html#483" class="Module">F-Coalgebra</a> <a id="1908" href="Categories.Object.Terminal.html#905" class="Function">⊤</a> <a id="1910" class="Keyword">using</a> <a id="1916" class="Symbol">()</a> <a id="1919" class="Keyword">renaming</a> <a id="1928" class="Symbol">(</a><a id="1929" href="Categories.Functor.Coalgebra.html#571" class="Field">A</a> <a id="1931" class="Symbol">to</a> <a id="1934" class="Field">DX</a><a id="1936" class="Symbol">)</a> <a id="1938" class="Keyword">public</a>
<a id="1952" href="Monad.Instance.Delay.html#1952" class="Function">out-≅</a> <a id="1958" class="Symbol">:</a> <a id="1960" href="Monad.Instance.Delay.html#1934" class="Function">DX</a> <a id="1963" href="Categories.Morphism.html#1958" class="Record Operator">≅</a> <a id="1965" href="Monad.Instance.Delay.html#1812" class="Bound">X</a> <a id="1967" href="Categories.Category.Cocartesian.html#1549" class="Function Operator">+</a> <a id="1969" href="Monad.Instance.Delay.html#1934" class="Function">DX</a>
<a id="1978" href="Monad.Instance.Delay.html#1952" class="Function">out-≅</a> <a id="1984" class="Symbol">=</a> <a id="1986" href="Categories.Category.Construction.F-Coalgebras.html#3360" class="Function">colambek</a> <a id="1995" class="Symbol">{</a><a id="1996" class="Argument">F</a> <a id="1998" class="Symbol">=</a> <a id="2000" href="Monad.Instance.Delay.html#1812" class="Bound">X</a> <a id="2002" href="Categories.Category.Cocartesian.html#3335" class="Function Operator">+-</a> <a id="2005" class="Symbol">}</a> <a id="2007" class="Symbol">(</a><a id="2008" href="Monad.Instance.Delay.html#1739" class="Field">coalgebras</a> <a id="2019" href="Monad.Instance.Delay.html#1812" class="Bound">X</a><a id="2020" class="Symbol">)</a>
<a id="2029" class="Comment">-- note: out-≅.from ≡ ⊤.α</a>
<a id="2061" class="Keyword">open</a> <a id="2066" href="Categories.Morphism.html#1958" class="Module Operator">_≅_</a> <a id="2070" href="Monad.Instance.Delay.html#1952" class="Function">out-≅</a> <a id="2076" class="Keyword">using</a> <a id="2082" class="Symbol">()</a> <a id="2085" class="Keyword">renaming</a> <a id="2094" class="Symbol">(</a><a id="2095" href="Categories.Morphism.html#2023" class="Field">to</a> <a id="2098" class="Symbol">to</a> <a id="2101" class="Field">out⁻¹</a><a id="2106" class="Symbol">;</a> <a id="2108" href="Categories.Morphism.html#2006" class="Field">from</a> <a id="2113" class="Symbol">to</a> <a id="2116" class="Field">out</a><a id="2119" class="Symbol">)</a> <a id="2121" class="Keyword">public</a>
<a id="2135" href="Monad.Instance.Delay.html#2135" class="Function">now</a> <a id="2139" class="Symbol">:</a> <a id="2141" href="Monad.Instance.Delay.html#1812" class="Bound">X</a> <a id="2143" href="Categories.Category.Core.html#575" class="Function Operator">⇒</a> <a id="2145" href="Monad.Instance.Delay.html#1934" class="Function">DX</a>
<a id="2154" href="Monad.Instance.Delay.html#2135" class="Function">now</a> <a id="2158" class="Symbol">=</a> <a id="2160" href="Monad.Instance.Delay.html#2101" class="Function">out⁻¹</a> <a id="2166" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="2168" href="Categories.Object.Coproduct.html#492" class="Function">i₁</a>
<a id="2178" href="Monad.Instance.Delay.html#2178" class="Function">later</a> <a id="2184" class="Symbol">:</a> <a id="2186" href="Monad.Instance.Delay.html#1934" class="Function">DX</a> <a id="2189" href="Categories.Category.Core.html#575" class="Function Operator">⇒</a> <a id="2191" href="Monad.Instance.Delay.html#1934" class="Function">DX</a>
<a id="2200" href="Monad.Instance.Delay.html#2178" class="Function">later</a> <a id="2206" class="Symbol">=</a> <a id="2208" href="Monad.Instance.Delay.html#2101" class="Function">out⁻¹</a> <a id="2214" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="2216" href="Categories.Object.Coproduct.html#512" class="Function">i₂</a>
<a id="2226" class="Comment">-- convenient notation</a>
<a id="2255" href="Monad.Instance.Delay.html#2255" class="Function">▷</a> <a id="2257" class="Symbol">=</a> <a id="2259" href="Monad.Instance.Delay.html#2178" class="Function">later</a>
<a id="2272" href="Monad.Instance.Delay.html#2272" class="Function">unitlaw</a> <a id="2280" class="Symbol">:</a> <a id="2282" href="Monad.Instance.Delay.html#2116" class="Function">out</a> <a id="2286" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="2288" href="Monad.Instance.Delay.html#2135" class="Function">now</a> <a id="2292" href="Categories.Category.Core.html#595" class="Function Operator">≈</a> <a id="2294" href="Categories.Object.Coproduct.html#492" class="Function">i₁</a>
<a id="2303" href="Monad.Instance.Delay.html#2272" class="Function">unitlaw</a> <a id="2311" class="Symbol">=</a> <a id="2313" href="Categories.Morphism.Reasoning.Core.html#6851" class="Function">cancelˡ</a> <a id="2321" class="Symbol">(</a><a id="2322" href="Categories.Morphism.html#1612" class="Function">_≅_.isoʳ</a> <a id="2331" href="Monad.Instance.Delay.html#1952" class="Function">out-≅</a><a id="2336" class="Symbol">)</a>
<a id="2345" href="Monad.Instance.Delay.html#2345" class="Function">laterlaw</a> <a id="2354" class="Symbol">:</a> <a id="2356" href="Monad.Instance.Delay.html#2116" class="Function">out</a> <a id="2360" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="2362" href="Monad.Instance.Delay.html#2178" class="Function">later</a> <a id="2368" href="Categories.Category.Core.html#595" class="Function Operator">≈</a> <a id="2370" href="Categories.Object.Coproduct.html#512" class="Function">i₂</a>
<a id="2379" href="Monad.Instance.Delay.html#2345" class="Function">laterlaw</a> <a id="2388" class="Symbol">=</a> <a id="2390" href="Categories.Morphism.Reasoning.Core.html#6851" class="Function">cancelˡ</a> <a id="2398" class="Symbol">(</a><a id="2399" href="Categories.Morphism.html#1612" class="Function">_≅_.isoʳ</a> <a id="2408" href="Monad.Instance.Delay.html#1952" class="Function">out-≅</a><a id="2413" class="Symbol">)</a>
</pre>
<p>Since <code>Y ⇒ X + Y</code> is an algebra for the
<code>(X + -)</code> functor, we can use the final coalgebras to get a
<em>coiteration operator</em> <code>coit</code>:</p>
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<pre class="Agda"> <a id="3254" class="Keyword">module</a> <a id="3261" href="Monad.Instance.Delay.html#3261" class="Module">_</a> <a id="3263" class="Symbol">{</a><a id="3264" href="Monad.Instance.Delay.html#3264" class="Bound">Y</a> <a id="3266" class="Symbol">:</a> <a id="3268" href="Categories.Category.Core.html#559" class="Function">Obj</a><a id="3271" class="Symbol">}</a> <a id="3273" class="Keyword">where</a>
<a id="3288" href="Monad.Instance.Delay.html#3288" class="Function">coit</a> <a id="3293" class="Symbol">:</a> <a id="3295" href="Monad.Instance.Delay.html#3264" class="Bound">Y</a> <a id="3297" href="Categories.Category.Core.html#575" class="Function Operator">⇒</a> <a id="3299" href="Monad.Instance.Delay.html#1812" class="Bound">X</a> <a id="3301" href="Categories.Category.Cocartesian.html#1549" class="Function Operator">+</a> <a id="3303" href="Monad.Instance.Delay.html#3264" class="Bound">Y</a> <a id="3305" class="Symbol">→</a> <a id="3307" href="Monad.Instance.Delay.html#3264" class="Bound">Y</a> <a id="3309" href="Categories.Category.Core.html#575" class="Function Operator">⇒</a> <a id="3311" href="Monad.Instance.Delay.html#1934" class="Function">DX</a>
<a id="3322" href="Monad.Instance.Delay.html#3288" class="Function">coit</a> <a id="3327" href="Monad.Instance.Delay.html#3327" class="Bound">f</a> <a id="3329" class="Symbol">=</a> <a id="3331" href="Monad.Instance.Delay.html#1644" class="Field">u</a> <a id="3333" class="Symbol">(</a><a id="3334" href="Categories.Object.Terminal.html#577" class="Function">!</a> <a id="3336" class="Symbol">{</a><a id="3337" class="Argument">A</a> <a id="3339" class="Symbol">=</a> <a id="3341" class="Keyword">record</a> <a id="3348" class="Symbol">{</a> <a id="3350" href="Categories.Functor.Coalgebra.html#571" class="Field">A</a> <a id="3352" class="Symbol">=</a> <a id="3354" href="Monad.Instance.Delay.html#3264" class="Bound">Y</a> <a id="3356" class="Symbol">;</a> <a id="3358" href="Categories.Functor.Coalgebra.html#585" class="Field">α</a> <a id="3360" class="Symbol">=</a> <a id="3362" href="Monad.Instance.Delay.html#3327" class="Bound">f</a> <a id="3364" class="Symbol">}})</a>
<a id="3377" href="Monad.Instance.Delay.html#3377" class="Function">coit-commutes</a> <a id="3391" class="Symbol">:</a> <a id="3393" class="Symbol">∀</a> <a id="3395" class="Symbol">(</a><a id="3396" href="Monad.Instance.Delay.html#3396" class="Bound">f</a> <a id="3398" class="Symbol">:</a> <a id="3400" href="Monad.Instance.Delay.html#3264" class="Bound">Y</a> <a id="3402" href="Categories.Category.Core.html#575" class="Function Operator">⇒</a> <a id="3404" href="Monad.Instance.Delay.html#1812" class="Bound">X</a> <a id="3406" href="Categories.Category.Cocartesian.html#1549" class="Function Operator">+</a> <a id="3408" href="Monad.Instance.Delay.html#3264" class="Bound">Y</a><a id="3409" class="Symbol">)</a> <a id="3411" class="Symbol">→</a> <a id="3413" href="Monad.Instance.Delay.html#2116" class="Function">out</a> <a id="3417" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="3419" class="Symbol">(</a><a id="3420" href="Monad.Instance.Delay.html#3288" class="Function">coit</a> <a id="3425" href="Monad.Instance.Delay.html#3396" class="Bound">f</a><a id="3426" class="Symbol">)</a> <a id="3428" href="Categories.Category.Core.html#595" class="Function Operator">≈</a> <a id="3430" class="Symbol">(</a><a id="3431" href="Category.Ambient.html#2115" class="Function">idC</a> <a id="3435" href="Categories.Category.Cocartesian.html#2133" class="Function Operator">+₁</a> <a id="3438" href="Monad.Instance.Delay.html#3288" class="Function">coit</a> <a id="3443" href="Monad.Instance.Delay.html#3396" class="Bound">f</a><a id="3444" class="Symbol">)</a> <a id="3446" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="3448" href="Monad.Instance.Delay.html#3396" class="Bound">f</a>
<a id="3458" href="Monad.Instance.Delay.html#3377" class="Function">coit-commutes</a> <a id="3472" href="Monad.Instance.Delay.html#3472" class="Bound">f</a> <a id="3474" class="Symbol">=</a> <a id="3476" href="Categories.Functor.Coalgebra.html#1433" class="Field">commutes</a> <a id="3485" class="Symbol">(</a><a id="3486" href="Categories.Object.Terminal.html#577" class="Function">!</a> <a id="3488" class="Symbol">{</a><a id="3489" class="Argument">A</a> <a id="3491" class="Symbol">=</a> <a id="3493" class="Keyword">record</a> <a id="3500" class="Symbol">{</a> <a id="3502" href="Categories.Functor.Coalgebra.html#571" class="Field">A</a> <a id="3504" class="Symbol">=</a> <a id="3506" href="Monad.Instance.Delay.html#3264" class="Bound">Y</a> <a id="3508" class="Symbol">;</a> <a id="3510" href="Categories.Functor.Coalgebra.html#585" class="Field">α</a> <a id="3512" class="Symbol">=</a> <a id="3514" href="Monad.Instance.Delay.html#3472" class="Bound">f</a> <a id="3516" class="Symbol">}})</a>
<a id="3529" href="Monad.Instance.Delay.html#3529" class="Function">coit-resp-≈</a> <a id="3541" class="Symbol">:</a> <a id="3543" class="Symbol">∀</a> <a id="3545" class="Symbol">{</a><a id="3546" href="Monad.Instance.Delay.html#3546" class="Bound">f</a> <a id="3548" href="Monad.Instance.Delay.html#3548" class="Bound">g</a> <a id="3550" class="Symbol">:</a> <a id="3552" href="Monad.Instance.Delay.html#3264" class="Bound">Y</a> <a id="3554" href="Categories.Category.Core.html#575" class="Function Operator">⇒</a> <a id="3556" href="Monad.Instance.Delay.html#1812" class="Bound">X</a> <a id="3558" href="Categories.Category.Cocartesian.html#1549" class="Function Operator">+</a> <a id="3560" href="Monad.Instance.Delay.html#3264" class="Bound">Y</a><a id="3561" class="Symbol">}</a> <a id="3563" class="Symbol">→</a> <a id="3565" href="Monad.Instance.Delay.html#3546" class="Bound">f</a> <a id="3567" href="Categories.Category.Core.html#595" class="Function Operator">≈</a> <a id="3569" href="Monad.Instance.Delay.html#3548" class="Bound">g</a> <a id="3571" class="Symbol">→</a> <a id="3573" href="Monad.Instance.Delay.html#3288" class="Function">coit</a> <a id="3578" href="Monad.Instance.Delay.html#3546" class="Bound">f</a> <a id="3580" href="Categories.Category.Core.html#595" class="Function Operator">≈</a> <a id="3582" href="Monad.Instance.Delay.html#3288" class="Function">coit</a> <a id="3587" href="Monad.Instance.Delay.html#3548" class="Bound">g</a>
<a id="3597" href="Monad.Instance.Delay.html#3529" class="Function">coit-resp-≈</a> <a id="3609" class="Symbol">{</a><a id="3610" href="Monad.Instance.Delay.html#3610" class="Bound">f</a><a id="3611" class="Symbol">}</a> <a id="3613" class="Symbol">{</a><a id="3614" href="Monad.Instance.Delay.html#3614" class="Bound">g</a><a id="3615" class="Symbol">}</a> <a id="3617" href="Monad.Instance.Delay.html#3617" class="Bound">eq</a> <a id="3620" class="Symbol">=</a> <a id="3622" href="Categories.Object.Terminal.html#605" class="Function">Terminal.!-unique</a> <a id="3640" class="Symbol">(</a><a id="3641" href="Monad.Instance.Delay.html#1739" class="Field">coalgebras</a> <a id="3652" href="Monad.Instance.Delay.html#1812" class="Bound">X</a><a id="3653" class="Symbol">)</a> <a id="3655" class="Symbol">(</a><a id="3656" class="Keyword">record</a> <a id="3663" class="Symbol">{</a> <a id="3665" href="Categories.Functor.Coalgebra.html#1411" class="Field">f</a> <a id="3667" class="Symbol">=</a> <a id="3669" href="Monad.Instance.Delay.html#3288" class="Function">coit</a> <a id="3674" href="Monad.Instance.Delay.html#3614" class="Bound">g</a> <a id="3676" class="Symbol">;</a> <a id="3678" href="Categories.Functor.Coalgebra.html#1433" class="Field">commutes</a> <a id="3687" class="Symbol">=</a> <a id="3689" href="Relation.Binary.Reasoning.Syntax.html#1510" class="Function Operator">begin</a>
<a id="3706" href="Monad.Instance.Delay.html#2116" class="Function">out</a> <a id="3710" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="3712" href="Monad.Instance.Delay.html#3288" class="Function">coit</a> <a id="3717" href="Monad.Instance.Delay.html#3614" class="Bound">g</a> <a id="3719" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="3722" href="Monad.Instance.Delay.html#3377" class="Function">coit-commutes</a> <a id="3736" href="Monad.Instance.Delay.html#3614" class="Bound">g</a> <a id="3738" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
<a id="3751" class="Symbol">(</a><a id="3752" href="Category.Ambient.html#2115" class="Function">idC</a> <a id="3756" href="Categories.Category.Cocartesian.html#2133" class="Function Operator">+₁</a> <a id="3759" href="Monad.Instance.Delay.html#3288" class="Function">coit</a> <a id="3764" href="Monad.Instance.Delay.html#3614" class="Bound">g</a><a id="3765" class="Symbol">)</a> <a id="3767" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="3769" href="Monad.Instance.Delay.html#3614" class="Bound">g</a> <a id="3771" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="3774" class="Symbol">(</a><a id="3775" href="Categories.Category.Core.html#2734" class="Function Operator">refl⟩∘⟨</a> <a id="3783" href="Relation.Binary.Structures.html#1622" class="Function">sym</a> <a id="3787" href="Monad.Instance.Delay.html#3617" class="Bound">eq</a><a id="3789" class="Symbol">)</a> <a id="3791" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
<a id="3804" class="Symbol">(</a><a id="3805" href="Category.Ambient.html#2115" class="Function">idC</a> <a id="3809" href="Categories.Category.Cocartesian.html#2133" class="Function Operator">+₁</a> <a id="3812" href="Monad.Instance.Delay.html#3288" class="Function">coit</a> <a id="3817" href="Monad.Instance.Delay.html#3614" class="Bound">g</a><a id="3818" class="Symbol">)</a> <a id="3820" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="3822" href="Monad.Instance.Delay.html#3610" class="Bound">f</a> <a id="3824" href="Relation.Binary.Reasoning.Syntax.html#12283" class="Function Operator">∎</a> <a id="3826" class="Symbol">})</a>
</pre>
<p>Furthermore if we combine the internal algebra structure of
Parametrized NNOs with Lambek’s lemma, we get the isomorphism
<code>X × N ≅ X + X × N</code>. At the same time the morphism
<code>X × N ⇒ X + X × N</code> is a coalgebra for the
<code>(Y + \_)</code>-functor defined above, this gives us another
morphism <code>ι : X × N ⇒ DX</code> by using the final coalgebras.</p>
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<pre class="Agda"> <a id="4993" href="Monad.Instance.Delay.html#4993" class="Function">nno-iso</a> <a id="5001" class="Symbol">:</a> <a id="5003" href="Monad.Instance.Delay.html#1812" class="Bound">X</a> <a id="5005" href="Categories.Category.BinaryProducts.html#1053" class="Function Operator">×</a> <a id="5007" href="Categories.Object.NaturalNumbers.Parametrized.html#2928" class="Function">N</a> <a id="5009" href="Categories.Morphism.html#1958" class="Record Operator">≅</a> <a id="5011" href="Monad.Instance.Delay.html#1812" class="Bound">X</a> <a id="5013" href="Categories.Category.Cocartesian.html#1549" class="Function Operator">+</a> <a id="5015" href="Monad.Instance.Delay.html#1812" class="Bound">X</a> <a id="5017" href="Categories.Category.BinaryProducts.html#1053" class="Function Operator">×</a> <a id="5019" href="Categories.Object.NaturalNumbers.Parametrized.html#2928" class="Function">N</a>
<a id="5027" href="Monad.Instance.Delay.html#4993" class="Function">nno-iso</a> <a id="5035" class="Symbol">=</a> <a id="5037" href="Categories.Category.Construction.F-Algebras.html#2829" class="Function">Lambek.lambek</a> <a id="5051" class="Symbol">(</a><a id="5052" class="Keyword">record</a> <a id="5059" class="Symbol">{</a> <a id="5061" href="Categories.Object.Initial.html#804" class="Field">⊥</a> <a id="5063" class="Symbol">=</a> <a id="5065" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#6310" class="Function">PNNO-Algebra</a> <a id="5078" href="Category.Ambient.html#1969" class="Function">cartesianCategory</a> <a id="5096" href="Categories.Category.Cocartesian.html#3462" class="Function">coproducts</a> <a id="5107" href="Monad.Instance.Delay.html#1812" class="Bound">X</a> <a id="5109" href="Categories.Object.NaturalNumbers.Parametrized.html#2928" class="Function">N</a> <a id="5111" href="Categories.Object.NaturalNumbers.Parametrized.html#960" class="Function">z</a> <a id="5113" href="Categories.Object.NaturalNumbers.Parametrized.html#974" class="Function">s</a> <a id="5115" class="Symbol">;</a> <a id="5117" href="Categories.Object.Initial.html#816" class="Field">⊥-is-initial</a> <a id="5130" class="Symbol">=</a> <a id="5132" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#9333" class="Function">PNNO⇒Initial₂</a> <a id="5146" href="Category.Ambient.html#1969" class="Function">cartesianCategory</a> <a id="5164" href="Categories.Category.Cocartesian.html#3462" class="Function">coproducts</a> <a id="5175" href="Category.Ambient.html#1842" class="Field">ℕ</a> <a id="5177" href="Monad.Instance.Delay.html#1812" class="Bound">X</a> <a id="5179" class="Symbol">})</a>
<a id="5189" href="Monad.Instance.Delay.html#5189" class="Function">ι-coalg</a> <a id="5197" class="Symbol">:</a> <a id="5199" href="Categories.Functor.Coalgebra.html#1350" class="Record">F-Coalgebra-Morphism</a> <a id="5220" class="Symbol">(</a><a id="5221" class="Keyword">record</a> <a id="5228" class="Symbol">{</a> <a id="5230" href="Categories.Functor.Coalgebra.html#571" class="Field">A</a> <a id="5232" class="Symbol">=</a> <a id="5234" href="Monad.Instance.Delay.html#1812" class="Bound">X</a> <a id="5236" href="Categories.Category.BinaryProducts.html#1053" class="Function Operator">×</a> <a id="5238" href="Categories.Object.NaturalNumbers.Parametrized.html#2928" class="Function">N</a> <a id="5240" class="Symbol">;</a> <a id="5242" href="Categories.Functor.Coalgebra.html#585" class="Field">α</a> <a id="5244" class="Symbol">=</a> <a id="5246" href="Categories.Morphism.html#2006" class="Field">_≅_.from</a> <a id="5255" href="Monad.Instance.Delay.html#4993" class="Function">nno-iso</a> <a id="5263" class="Symbol">})</a> <a id="5266" class="Symbol">(</a><a id="5267" class="Keyword">record</a> <a id="5274" class="Symbol">{</a> <a id="5276" href="Categories.Functor.Coalgebra.html#571" class="Field">A</a> <a id="5278" class="Symbol">=</a> <a id="5280" href="Monad.Instance.Delay.html#1934" class="Function">DX</a> <a id="5283" class="Symbol">;</a> <a id="5285" href="Categories.Functor.Coalgebra.html#585" class="Field">α</a> <a id="5287" class="Symbol">=</a> <a id="5289" href="Monad.Instance.Delay.html#2116" class="Function">out</a> <a id="5293" class="Symbol">})</a>
<a id="5302" href="Monad.Instance.Delay.html#5189" class="Function">ι-coalg</a> <a id="5310" class="Symbol">=</a> <a id="5312" href="Categories.Object.Terminal.html#577" class="Function">!</a> <a id="5314" class="Symbol">{</a><a id="5315" class="Argument">A</a> <a id="5317" class="Symbol">=</a> <a id="5319" class="Keyword">record</a> <a id="5326" class="Symbol">{</a> <a id="5328" href="Categories.Functor.Coalgebra.html#571" class="Field">A</a> <a id="5330" class="Symbol">=</a> <a id="5332" href="Monad.Instance.Delay.html#1812" class="Bound">X</a> <a id="5334" href="Categories.Category.BinaryProducts.html#1053" class="Function Operator">×</a> <a id="5336" href="Categories.Object.NaturalNumbers.Parametrized.html#2928" class="Function">N</a> <a id="5338" class="Symbol">;</a> <a id="5340" href="Categories.Functor.Coalgebra.html#585" class="Field">α</a> <a id="5342" class="Symbol">=</a> <a id="5344" href="Categories.Morphism.html#2006" class="Field">_≅_.from</a> <a id="5353" href="Monad.Instance.Delay.html#4993" class="Function">nno-iso</a> <a id="5361" class="Symbol">}}</a>
<a id="5371" href="Monad.Instance.Delay.html#5371" class="Function">ι</a> <a id="5373" class="Symbol">:</a> <a id="5375" href="Monad.Instance.Delay.html#1812" class="Bound">X</a> <a id="5377" href="Categories.Category.BinaryProducts.html#1053" class="Function Operator">×</a> <a id="5379" href="Categories.Object.NaturalNumbers.Parametrized.html#2928" class="Function">N</a> <a id="5381" href="Categories.Category.Core.html#575" class="Function Operator">⇒</a> <a id="5383" href="Monad.Instance.Delay.html#1934" class="Function">DX</a>
<a id="5392" href="Monad.Instance.Delay.html#5371" class="Function">ι</a> <a id="5394" class="Symbol">=</a> <a id="5396" href="Monad.Instance.Delay.html#1644" class="Field">u</a> <a id="5398" href="Monad.Instance.Delay.html#5189" class="Function">ι-coalg</a>
<a id="5413" href="Monad.Instance.Delay.html#5413" class="Function">ι-commutes</a> <a id="5424" class="Symbol">:</a> <a id="5426" href="Monad.Instance.Delay.html#2116" class="Function">out</a> <a id="5430" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="5432" href="Monad.Instance.Delay.html#5371" class="Function">ι</a> <a id="5434" href="Categories.Category.Core.html#595" class="Function Operator">≈</a> <a id="5436" class="Symbol">(</a><a id="5437" href="Category.Ambient.html#2115" class="Function">idC</a> <a id="5441" href="Categories.Category.Cocartesian.html#2133" class="Function Operator">+₁</a> <a id="5444" href="Monad.Instance.Delay.html#5371" class="Function">ι</a><a id="5445" class="Symbol">)</a> <a id="5447" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="5449" href="Categories.Morphism.html#2006" class="Field">_≅_.from</a> <a id="5458" href="Monad.Instance.Delay.html#4993" class="Function">nno-iso</a>
<a id="5472" href="Monad.Instance.Delay.html#5413" class="Function">ι-commutes</a> <a id="5483" class="Symbol">=</a> <a id="5485" href="Categories.Functor.Coalgebra.html#1433" class="Field">commutes</a> <a id="5494" href="Monad.Instance.Delay.html#5189" class="Function">ι-coalg</a>
</pre>
<h2 id="delay-is-a-monad">Delay is a monad</h2>
<p>The next step is showing that this actually yields a monad. Some
parts for this are already given, we can construct <code>D X</code> from
<code>X</code> and <code>now : D X ⇒ D X</code> is the monad unit.
What’s missing is Kleisli-lifting, given a morphism
<code>f : X ⇒ D Y</code> we need to construct a morphism
<code>extend f : D X ⇒ D Y</code>. To do so we go from <code>D X</code>
to <code>D X + D Y</code> via injection and then construct a coalgebra
<code>D X + D Y ⇒ Y + (D X + D Y)</code>, the final coalgebra
<code>D Y ⇒ Y + D Y</code> then yields a coalgebra-morphism from
<code>D X + D Y</code> to <code>D Y</code>, see the following
diagram:</p>
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<p>Proving the monad laws then requires two key lemmas, first of all
that the following diagram commutes for any <code>f</code> (this is
<code>extendlaw</code>)</p>
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<p>and second that <code>extend f</code> is the unique morphism
satisfying the commutative diagram i.e. any other morphism for which the
diagram commutes must be equivalent to <code>extend f</code> (this is
<code>extend-unique</code>).</p>
<pre class="Agda"> <a id="7734" class="Comment">-- Make DX more accessible outside</a>
<a id="DelayM.D₀"></a><a id="7773" href="Monad.Instance.Delay.html#7773" class="Function">D₀</a> <a id="7776" class="Symbol">:</a> <a id="7778" href="Categories.Category.Core.html#559" class="Function">Obj</a> <a id="7782" class="Symbol">→</a> <a id="7784" href="Categories.Category.Core.html#559" class="Function">Obj</a>
<a id="7792" href="Monad.Instance.Delay.html#7773" class="Function">D₀</a> <a id="7795" href="Monad.Instance.Delay.html#7795" class="Bound">X</a> <a id="7797" class="Symbol">=</a> <a id="7799" href="Monad.Instance.Delay.html#1934" class="Function">DX</a> <a id="7802" class="Symbol">{</a><a id="7803" href="Monad.Instance.Delay.html#7795" class="Bound">X</a><a id="7804" class="Symbol">}</a>
<a id="7811" class="Keyword">module</a> <a id="7818" href="Monad.Instance.Delay.html#7818" class="Module">_</a> <a id="7820" class="Symbol">{</a><a id="7821" href="Monad.Instance.Delay.html#7821" class="Bound">X</a> <a id="7823" href="Monad.Instance.Delay.html#7823" class="Bound">Y</a> <a id="7825" class="Symbol">:</a> <a id="7827" href="Categories.Category.Core.html#559" class="Function">Obj</a><a id="7830" class="Symbol">}</a> <a id="7832" class="Symbol">(</a><a id="7833" href="Monad.Instance.Delay.html#7833" class="Bound">f</a> <a id="7835" class="Symbol">:</a> <a id="7837" href="Monad.Instance.Delay.html#7821" class="Bound">X</a> <a id="7839" href="Categories.Category.Core.html#575" class="Function Operator">⇒</a> <a id="7841" href="Monad.Instance.Delay.html#7773" class="Function">D₀</a> <a id="7844" href="Monad.Instance.Delay.html#7823" class="Bound">Y</a><a id="7845" class="Symbol">)</a> <a id="7847" class="Keyword">where</a>
<a id="7859" class="Keyword">open</a> <a id="7864" href="Categories.Object.Terminal.html#860" class="Module">Terminal</a>
<a id="7879" href="Monad.Instance.Delay.html#7879" class="Function">alg</a> <a id="7883" class="Symbol">:</a> <a id="7885" href="Categories.Functor.Coalgebra.html#483" class="Record">F-Coalgebra</a> <a id="7897" class="Symbol">(</a><a id="7898" href="Monad.Instance.Delay.html#7823" class="Bound">Y</a> <a id="7900" href="Categories.Category.Cocartesian.html#3335" class="Function Operator">+-</a><a id="7902" class="Symbol">)</a>
<a id="7910" href="Monad.Instance.Delay.html#7879" class="Function">alg</a> <a id="7914" class="Symbol">=</a> <a id="7916" class="Keyword">record</a> <a id="7923" class="Symbol">{</a> <a id="7925" href="Categories.Functor.Coalgebra.html#571" class="Field">A</a> <a id="7927" class="Symbol">=</a> <a id="7929" href="Monad.Instance.Delay.html#7773" class="Function">D₀</a> <a id="7932" href="Monad.Instance.Delay.html#7821" class="Bound">X</a> <a id="7934" href="Categories.Category.Cocartesian.html#1549" class="Function Operator">+</a> <a id="7936" href="Monad.Instance.Delay.html#7773" class="Function">D₀</a> <a id="7939" href="Monad.Instance.Delay.html#7823" class="Bound">Y</a> <a id="7941" class="Symbol">;</a> <a id="7943" href="Categories.Functor.Coalgebra.html#585" class="Field">α</a> <a id="7945" class="Symbol">=</a> <a id="7947" href="Categories.Object.Coproduct.html#532" class="Function Operator">[</a> <a id="7949" href="Categories.Object.Coproduct.html#532" class="Function Operator">[</a> <a id="7951" href="Categories.Object.Coproduct.html#532" class="Function Operator">[</a> <a id="7953" href="Categories.Object.Coproduct.html#492" class="Function">i₁</a> <a id="7956" href="Categories.Object.Coproduct.html#532" class="Function Operator">,</a> <a id="7958" href="Categories.Object.Coproduct.html#512" class="Function">i₂</a> <a id="7961" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="7963" href="Categories.Object.Coproduct.html#512" class="Function">i₂</a> <a id="7966" href="Categories.Object.Coproduct.html#532" class="Function Operator">]</a> <a id="7968" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="7970" class="Symbol">(</a><a id="7971" href="Monad.Instance.Delay.html#2116" class="Function">out</a> <a id="7975" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="7977" href="Monad.Instance.Delay.html#7833" class="Bound">f</a><a id="7978" class="Symbol">)</a> <a id="7980" href="Categories.Object.Coproduct.html#532" class="Function Operator">,</a> <a id="7982" href="Categories.Object.Coproduct.html#512" class="Function">i₂</a> <a id="7985" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="7987" href="Categories.Object.Coproduct.html#492" class="Function">i₁</a> <a id="7990" href="Categories.Object.Coproduct.html#532" class="Function Operator">]</a> <a id="7992" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="7994" href="Monad.Instance.Delay.html#2116" class="Function">out</a> <a id="7998" href="Categories.Object.Coproduct.html#532" class="Function Operator">,</a> <a id="8000" class="Symbol">(</a><a id="8001" href="Category.Ambient.html#2115" class="Function">idC</a> <a id="8005" href="Categories.Category.Cocartesian.html#2133" class="Function Operator">+₁</a> <a id="8008" href="Categories.Object.Coproduct.html#512" class="Function">i₂</a><a id="8010" class="Symbol">)</a> <a id="8012" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="8014" href="Monad.Instance.Delay.html#2116" class="Function">out</a> <a id="8018" href="Categories.Object.Coproduct.html#532" class="Function Operator">]</a> <a id="8020" class="Symbol">}</a>
<a id="8028" href="Monad.Instance.Delay.html#8028" class="Function">extend&#39;</a> <a id="8036" class="Symbol">:</a> <a id="8038" href="Monad.Instance.Delay.html#7773" class="Function">D₀</a> <a id="8041" href="Monad.Instance.Delay.html#7821" class="Bound">X</a> <a id="8043" href="Categories.Category.Core.html#575" class="Function Operator">⇒</a> <a id="8045" href="Monad.Instance.Delay.html#7773" class="Function">D₀</a> <a id="8048" href="Monad.Instance.Delay.html#7823" class="Bound">Y</a>
<a id="8056" href="Monad.Instance.Delay.html#8028" class="Function">extend&#39;</a> <a id="8064" class="Symbol">=</a> <a id="8066" href="Monad.Instance.Delay.html#1644" class="Field">u</a> <a id="8068" class="Symbol">(</a><a id="8069" href="Categories.Object.Terminal.html#577" class="Function">!</a> <a id="8071" class="Symbol">(</a><a id="8072" href="Monad.Instance.Delay.html#1739" class="Field">coalgebras</a> <a id="8083" href="Monad.Instance.Delay.html#7823" class="Bound">Y</a><a id="8084" class="Symbol">)</a> <a id="8086" class="Symbol">{</a><a id="8087" class="Argument">A</a> <a id="8089" class="Symbol">=</a> <a id="8091" href="Monad.Instance.Delay.html#7879" class="Function">alg</a><a id="8094" class="Symbol">})</a> <a id="8097" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="8099" href="Categories.Object.Coproduct.html#492" class="Function">i₁</a> <a id="8102" class="Symbol">{</a><a id="8103" class="Argument">B</a> <a id="8105" class="Symbol">=</a> <a id="8107" href="Monad.Instance.Delay.html#7773" class="Function">D₀</a> <a id="8110" href="Monad.Instance.Delay.html#7823" class="Bound">Y</a><a id="8111" class="Symbol">}</a>
<a id="8120" href="Monad.Instance.Delay.html#8120" class="Function">!∘i₂</a> <a id="8125" class="Symbol">:</a> <a id="8127" href="Monad.Instance.Delay.html#1644" class="Field">u</a> <a id="8129" class="Symbol">(</a><a id="8130" href="Categories.Object.Terminal.html#577" class="Function">!</a> <a id="8132" class="Symbol">(</a><a id="8133" href="Monad.Instance.Delay.html#1739" class="Field">coalgebras</a> <a id="8144" href="Monad.Instance.Delay.html#7823" class="Bound">Y</a><a id="8145" class="Symbol">)</a> <a id="8147" class="Symbol">{</a><a id="8148" class="Argument">A</a> <a id="8150" class="Symbol">=</a> <a id="8152" href="Monad.Instance.Delay.html#7879" class="Function">alg</a><a id="8155" class="Symbol">})</a> <a id="8158" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="8160" href="Categories.Object.Coproduct.html#512" class="Function">i₂</a> <a id="8163" href="Categories.Category.Core.html#595" class="Function Operator">≈</a> <a id="8165" href="Category.Ambient.html#2115" class="Function">idC</a>
<a id="8175" href="Monad.Instance.Delay.html#8120" class="Function">!∘i₂</a> <a id="8180" class="Symbol">=</a> <a id="8182" href="Categories.Object.Terminal.html#803" class="Function">⊤-id</a> <a id="8187" class="Symbol">(</a><a id="8188" href="Monad.Instance.Delay.html#1739" class="Field">coalgebras</a> <a id="8199" href="Monad.Instance.Delay.html#7823" class="Bound">Y</a><a id="8200" class="Symbol">)</a> <a id="8202" class="Symbol">(</a><a id="8203" href="Categories.Category.Construction.F-Coalgebras.html#598" class="Function">F-Coalgebras</a> <a id="8216" class="Symbol">(</a><a id="8217" href="Monad.Instance.Delay.html#7823" class="Bound">Y</a> <a id="8219" href="Categories.Category.Cocartesian.html#3335" class="Function Operator">+-</a><a id="8221" class="Symbol">)</a> <a id="8223" href="Categories.Category.html#700" class="Function Operator">[</a> <a id="8225" href="Categories.Object.Terminal.html#577" class="Function">!</a> <a id="8227" class="Symbol">(</a><a id="8228" href="Monad.Instance.Delay.html#1739" class="Field">coalgebras</a> <a id="8239" href="Monad.Instance.Delay.html#7823" class="Bound">Y</a><a id="8240" class="Symbol">)</a> <a id="8242" href="Categories.Category.html#700" class="Function Operator">∘</a> <a id="8244" class="Keyword">record</a> <a id="8251" class="Symbol">{</a> <a id="8253" href="Categories.Functor.Coalgebra.html#1411" class="Field">f</a> <a id="8255" class="Symbol">=</a> <a id="8257" href="Categories.Object.Coproduct.html#512" class="Function">i₂</a> <a id="8260" class="Symbol">;</a> <a id="8262" href="Categories.Functor.Coalgebra.html#1433" class="Field">commutes</a> <a id="8271" class="Symbol">=</a> <a id="8273" href="Categories.Object.Coproduct.html#602" class="Function">inject₂</a> <a id="8281" class="Symbol">}</a> <a id="8283" href="Categories.Category.html#700" class="Function Operator">]</a> <a id="8285" class="Symbol">)</a>
<a id="8294" href="Monad.Instance.Delay.html#8294" class="Function">extendlaw</a> <a id="8304" class="Symbol">:</a> <a id="8306" href="Monad.Instance.Delay.html#2116" class="Function">out</a> <a id="8310" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="8312" href="Monad.Instance.Delay.html#8028" class="Function">extend&#39;</a> <a id="8320" href="Categories.Category.Core.html#595" class="Function Operator">≈</a> <a id="8322" href="Categories.Object.Coproduct.html#532" class="Function Operator">[</a> <a id="8324" href="Monad.Instance.Delay.html#2116" class="Function">out</a> <a id="8328" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="8330" href="Monad.Instance.Delay.html#7833" class="Bound">f</a> <a id="8332" href="Categories.Object.Coproduct.html#532" class="Function Operator">,</a> <a id="8334" href="Categories.Object.Coproduct.html#512" class="Function">i₂</a> <a id="8337" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="8339" href="Monad.Instance.Delay.html#8028" class="Function">extend&#39;</a> <a id="8347" href="Categories.Object.Coproduct.html#532" class="Function Operator">]</a> <a id="8349" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="8351" href="Monad.Instance.Delay.html#2116" class="Function">out</a>
<a id="8361" href="Monad.Instance.Delay.html#8294" class="Function">extendlaw</a> <a id="8371" class="Symbol">=</a> <a id="8373" href="Relation.Binary.Reasoning.Syntax.html#1510" class="Function Operator">begin</a>
<a id="8388" href="Monad.Instance.Delay.html#2116" class="Function">out</a> <a id="8392" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="8394" href="Monad.Instance.Delay.html#8028" class="Function">extend&#39;</a> <a id="8402" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="8405" href="Categories.Morphism.Reasoning.Core.html#2048" class="Function">pullˡ</a> <a id="8411" class="Symbol">(</a><a id="8412" href="Categories.Functor.Coalgebra.html#1433" class="Field">commutes</a> <a id="8421" class="Symbol">(</a><a id="8422" href="Categories.Object.Terminal.html#577" class="Function">!</a> <a id="8424" class="Symbol">(</a><a id="8425" href="Monad.Instance.Delay.html#1739" class="Field">coalgebras</a> <a id="8436" href="Monad.Instance.Delay.html#7823" class="Bound">Y</a><a id="8437" class="Symbol">)</a> <a id="8439" class="Symbol">{</a><a id="8440" class="Argument">A</a> <a id="8442" class="Symbol">=</a> <a id="8444" href="Monad.Instance.Delay.html#7879" class="Function">alg</a><a id="8447" class="Symbol">}))</a> <a id="8451" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
<a id="8462" class="Symbol">((</a><a id="8464" href="Category.Ambient.html#2115" class="Function">idC</a> <a id="8468" href="Categories.Category.Cocartesian.html#2133" class="Function Operator">+₁</a> <a id="8471" class="Symbol">(</a><a id="8472" href="Monad.Instance.Delay.html#1644" class="Field">u</a> <a id="8474" class="Symbol">(</a><a id="8475" href="Categories.Object.Terminal.html#577" class="Function">!</a> <a id="8477" class="Symbol">(</a><a id="8478" href="Monad.Instance.Delay.html#1739" class="Field">coalgebras</a> <a id="8489" href="Monad.Instance.Delay.html#7823" class="Bound">Y</a><a id="8490" class="Symbol">))))</a> <a id="8495" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="8497" href="Categories.Functor.Coalgebra.html#585" class="Field">α</a> <a id="8499" href="Monad.Instance.Delay.html#7879" class="Function">alg</a><a id="8502" class="Symbol">)</a> <a id="8504" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="8506" href="Categories.Object.Coproduct.html#492" class="Function">i₁</a> <a id="8516" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="8519" href="Categories.Morphism.Reasoning.Core.html#1914" class="Function">pullʳ</a> <a id="8525" href="Categories.Object.Coproduct.html#569" class="Function">inject₁</a> <a id="8533" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
<a id="8543" class="Symbol">(</a><a id="8544" href="Category.Ambient.html#2115" class="Function">idC</a> <a id="8548" href="Categories.Category.Cocartesian.html#2133" class="Function Operator">+₁</a> <a id="8551" class="Symbol">(</a><a id="8552" href="Monad.Instance.Delay.html#1644" class="Field">u</a> <a id="8554" class="Symbol">(</a><a id="8555" href="Categories.Object.Terminal.html#577" class="Function">!</a> <a id="8557" class="Symbol">(</a><a id="8558" href="Monad.Instance.Delay.html#1739" class="Field">coalgebras</a> <a id="8569" href="Monad.Instance.Delay.html#7823" class="Bound">Y</a><a id="8570" class="Symbol">))))</a> <a id="8575" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="8577" href="Categories.Object.Coproduct.html#532" class="Function Operator">[</a> <a id="8579" href="Categories.Object.Coproduct.html#532" class="Function Operator">[</a> <a id="8581" href="Categories.Object.Coproduct.html#492" class="Function">i₁</a> <a id="8584" href="Categories.Object.Coproduct.html#532" class="Function Operator">,</a> <a id="8586" href="Categories.Object.Coproduct.html#512" class="Function">i₂</a> <a id="8589" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="8591" href="Categories.Object.Coproduct.html#512" class="Function">i₂</a> <a id="8594" href="Categories.Object.Coproduct.html#532" class="Function Operator">]</a>
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<a id="12556" href="Categories.Object.Coproduct.html#532" class="Function Operator">[</a> <a id="12558" href="Categories.Object.Coproduct.html#532" class="Function Operator">[</a> <a id="12560" class="Symbol">(</a><a id="12561" href="Category.Ambient.html#2115" class="Function">idC</a> <a id="12565" href="Categories.Category.Cocartesian.html#2133" class="Function Operator">+₁</a> <a id="12568" href="Categories.Object.Coproduct.html#532" class="Function Operator">[</a> <a id="12570" href="Monad.Instance.Delay.html#9753" class="Bound">g</a> <a id="12572" href="Categories.Object.Coproduct.html#532" class="Function Operator">,</a> <a id="12574" href="Category.Ambient.html#2115" class="Function">idC</a> <a id="12578" href="Categories.Object.Coproduct.html#532" class="Function Operator">]</a><a id="12579" class="Symbol">)</a> <a id="12581" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="12583" href="Categories.Object.Coproduct.html#532" class="Function Operator">[</a> <a id="12585" href="Categories.Object.Coproduct.html#492" class="Function">i₁</a> <a id="12588" href="Categories.Object.Coproduct.html#532" class="Function Operator">,</a> <a id="12590" href="Categories.Object.Coproduct.html#512" class="Function">i₂</a> <a id="12593" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="12595" href="Categories.Object.Coproduct.html#512" class="Function">i₂</a> <a id="12598" href="Categories.Object.Coproduct.html#532" class="Function Operator">]</a> <a id="12600" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="12602" class="Symbol">(</a><a id="12603" href="Monad.Instance.Delay.html#2116" class="Function">out</a> <a id="12607" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="12609" href="Monad.Instance.Delay.html#7833" class="Bound">f</a><a id="12610" class="Symbol">)</a>
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<a id="12677" href="Categories.Object.Coproduct.html#532" class="Function Operator">,</a> <a id="12679" class="Symbol">(</a><a id="12680" href="Category.Ambient.html#2115" class="Function">idC</a> <a id="12684" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="12686" href="Category.Ambient.html#2115" class="Function">idC</a> <a id="12690" href="Categories.Category.Cocartesian.html#2133" class="Function Operator">+₁</a> <a id="12693" href="Categories.Object.Coproduct.html#532" class="Function Operator">[</a> <a id="12695" href="Monad.Instance.Delay.html#9753" class="Bound">g</a> <a id="12697" href="Categories.Object.Coproduct.html#532" class="Function Operator">,</a> <a id="12699" href="Category.Ambient.html#2115" class="Function">idC</a> <a id="12703" href="Categories.Object.Coproduct.html#532" class="Function Operator">]</a> <a id="12705" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="12707" href="Categories.Object.Coproduct.html#512" class="Function">i₂</a><a id="12709" class="Symbol">)</a> <a id="12711" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="12713" href="Monad.Instance.Delay.html#2116" class="Function">out</a> <a id="12717" href="Categories.Object.Coproduct.html#532" class="Function Operator">]</a> <a id="12736" href="Relation.Binary.Reasoning.Syntax.html#7400" class="Function">≈˘⟨</a> <a id="12740" href="Categories.Object.Coproduct.html#816" class="Function">[]-cong₂</a> <a id="12749" class="Symbol">(</a><a id="12750" href="Categories.Morphism.Reasoning.Core.html#2048" class="Function">pullˡ</a> <a id="12756" href="Categories.Category.Cocartesian.html#2736" class="Function">∘[]</a><a id="12759" class="Symbol">)</a> <a id="12761" class="Symbol">(</a><a id="12762" href="Categories.Morphism.Reasoning.Core.html#2048" class="Function">pullˡ</a> <a id="12768" href="Categories.Category.Cocartesian.html#2699" class="Function">+₁∘+₁</a><a id="12773" class="Symbol">)</a> <a id="12775" href="Relation.Binary.Reasoning.Syntax.html#7400" class="Function">⟩</a>
<a id="12787" href="Categories.Object.Coproduct.html#532" class="Function Operator">[</a> <a id="12789" class="Symbol">(</a><a id="12790" href="Category.Ambient.html#2115" class="Function">idC</a> <a id="12794" href="Categories.Category.Cocartesian.html#2133" class="Function Operator">+₁</a> <a id="12797" href="Categories.Object.Coproduct.html#532" class="Function Operator">[</a> <a id="12799" href="Monad.Instance.Delay.html#9753" class="Bound">g</a> <a id="12801" href="Categories.Object.Coproduct.html#532" class="Function Operator">,</a> <a id="12803" href="Category.Ambient.html#2115" class="Function">idC</a> <a id="12807" href="Categories.Object.Coproduct.html#532" class="Function Operator">]</a><a id="12808" class="Symbol">)</a> <a id="12810" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="12812" href="Categories.Object.Coproduct.html#532" class="Function Operator">[</a> <a id="12814" href="Categories.Object.Coproduct.html#532" class="Function Operator">[</a> <a id="12816" href="Categories.Object.Coproduct.html#492" class="Function">i₁</a> <a id="12819" href="Categories.Object.Coproduct.html#532" class="Function Operator">,</a> <a id="12821" href="Categories.Object.Coproduct.html#512" class="Function">i₂</a> <a id="12824" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="12826" href="Categories.Object.Coproduct.html#512" class="Function">i₂</a> <a id="12829" href="Categories.Object.Coproduct.html#532" class="Function Operator">]</a> <a id="12831" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="12833" class="Symbol">(</a><a id="12834" href="Monad.Instance.Delay.html#2116" class="Function">out</a> <a id="12838" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="12840" href="Monad.Instance.Delay.html#7833" class="Bound">f</a><a id="12841" class="Symbol">)</a> <a id="12843" href="Categories.Object.Coproduct.html#532" class="Function Operator">,</a> <a id="12845" href="Categories.Object.Coproduct.html#512" class="Function">i₂</a> <a id="12848" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="12850" href="Categories.Object.Coproduct.html#492" class="Function">i₁</a> <a id="12853" href="Categories.Object.Coproduct.html#532" class="Function Operator">]</a> <a id="12855" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="12857" href="Monad.Instance.Delay.html#2116" class="Function">out</a>
<a id="12872" href="Categories.Object.Coproduct.html#532" class="Function Operator">,</a> <a id="12874" class="Symbol">(</a><a id="12875" href="Category.Ambient.html#2115" class="Function">idC</a> <a id="12879" href="Categories.Category.Cocartesian.html#2133" class="Function Operator">+₁</a> <a id="12882" href="Categories.Object.Coproduct.html#532" class="Function Operator">[</a> <a id="12884" href="Monad.Instance.Delay.html#9753" class="Bound">g</a> <a id="12886" href="Categories.Object.Coproduct.html#532" class="Function Operator">,</a> <a id="12888" href="Category.Ambient.html#2115" class="Function">idC</a> <a id="12892" href="Categories.Object.Coproduct.html#532" class="Function Operator">]</a><a id="12893" class="Symbol">)</a> <a id="12895" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="12897" class="Symbol">(</a><a id="12898" href="Category.Ambient.html#2115" class="Function">idC</a> <a id="12902" href="Categories.Category.Cocartesian.html#2133" class="Function Operator">+₁</a> <a id="12905" href="Categories.Object.Coproduct.html#512" class="Function">i₂</a><a id="12907" class="Symbol">)</a> <a id="12909" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="12911" href="Monad.Instance.Delay.html#2116" class="Function">out</a> <a id="12915" href="Categories.Object.Coproduct.html#532" class="Function Operator">]</a> <a id="12931" href="Relation.Binary.Reasoning.Syntax.html#7400" class="Function">≈˘⟨</a> <a id="12935" href="Categories.Category.Cocartesian.html#2736" class="Function">∘[]</a> <a id="12939" href="Relation.Binary.Reasoning.Syntax.html#7400" class="Function">⟩</a>
<a id="12951" class="Symbol">(</a><a id="12952" href="Category.Ambient.html#2115" class="Function">idC</a> <a id="12956" href="Categories.Category.Cocartesian.html#2133" class="Function Operator">+₁</a> <a id="12959" href="Categories.Object.Coproduct.html#532" class="Function Operator">[</a> <a id="12961" href="Monad.Instance.Delay.html#9753" class="Bound">g</a> <a id="12963" href="Categories.Object.Coproduct.html#532" class="Function Operator">,</a> <a id="12965" href="Category.Ambient.html#2115" class="Function">idC</a> <a id="12969" href="Categories.Object.Coproduct.html#532" class="Function Operator">]</a><a id="12970" class="Symbol">)</a> <a id="12972" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="12974" href="Categories.Functor.Coalgebra.html#585" class="Field">α</a> <a id="12976" href="Monad.Instance.Delay.html#7879" class="Function">alg</a> <a id="12980" href="Relation.Binary.Reasoning.Syntax.html#12283" class="Function Operator">∎</a> <a id="12982" class="Symbol">}))</a> <a id="12986" href="Categories.Category.Core.html#2837" class="Function Operator">⟩∘⟨refl</a> <a id="12994" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
<a id="13005" href="Categories.Object.Coproduct.html#532" class="Function Operator">[</a> <a id="13007" href="Monad.Instance.Delay.html#9753" class="Bound">g</a> <a id="13009" href="Categories.Object.Coproduct.html#532" class="Function Operator">,</a> <a id="13011" href="Category.Ambient.html#2115" class="Function">idC</a> <a id="13015" href="Categories.Object.Coproduct.html#532" class="Function Operator">]</a> <a id="13017" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="13019" href="Categories.Object.Coproduct.html#492" class="Function">i₁</a> <a id="13068" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="13071" href="Categories.Object.Coproduct.html#569" class="Function">inject₁</a> <a id="13079" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
<a id="13089" href="Monad.Instance.Delay.html#9753" class="Bound">g</a> <a id="13091" href="Relation.Binary.Reasoning.Syntax.html#12283" class="Function Operator">∎</a>
<a id="13098" class="Comment">-- Lemma 39: ▷ ∘ f* ≈ (▷ ∘ f)* ≈ f* ∘ ▷</a>
<a id="13143" class="Keyword">module</a> <a id="13150" href="Monad.Instance.Delay.html#13150" class="Module">_</a> <a id="13152" class="Symbol">{</a><a id="13153" href="Monad.Instance.Delay.html#13153" class="Bound">X</a> <a id="13155" href="Monad.Instance.Delay.html#13155" class="Bound">Y</a><a id="13156" class="Symbol">}</a> <a id="13158" class="Symbol">(</a><a id="13159" href="Monad.Instance.Delay.html#13159" class="Bound">f</a> <a id="13161" class="Symbol">:</a> <a id="13163" href="Monad.Instance.Delay.html#13153" class="Bound">X</a> <a id="13165" href="Categories.Category.Core.html#575" class="Function Operator">⇒</a> <a id="13167" href="Monad.Instance.Delay.html#7773" class="Function">D₀</a> <a id="13170" href="Monad.Instance.Delay.html#13155" class="Bound">Y</a><a id="13171" class="Symbol">)</a> <a id="13173" class="Keyword">where</a>
<a id="13185" class="Keyword">private</a>
<a id="13203" href="Monad.Instance.Delay.html#13203" class="Function">helper</a> <a id="13210" class="Symbol">:</a> <a id="13212" href="Monad.Instance.Delay.html#2116" class="Function">out</a> <a id="13216" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="13218" href="Categories.Object.Coproduct.html#532" class="Function Operator">[</a> <a id="13220" href="Monad.Instance.Delay.html#13159" class="Bound">f</a> <a id="13222" href="Categories.Object.Coproduct.html#532" class="Function Operator">,</a> <a id="13224" href="Monad.Instance.Delay.html#8028" class="Function">extend&#39;</a> <a id="13232" class="Symbol">(</a><a id="13233" href="Monad.Instance.Delay.html#2255" class="Function">▷</a> <a id="13235" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="13237" href="Monad.Instance.Delay.html#13159" class="Bound">f</a><a id="13238" class="Symbol">)</a> <a id="13240" href="Categories.Object.Coproduct.html#532" class="Function Operator">]</a> <a id="13242" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="13244" href="Monad.Instance.Delay.html#2116" class="Function">out</a> <a id="13248" href="Categories.Category.Core.html#595" class="Function Operator">≈</a> <a id="13250" href="Categories.Object.Coproduct.html#532" class="Function Operator">[</a> <a id="13252" href="Monad.Instance.Delay.html#2116" class="Function">out</a> <a id="13256" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="13258" href="Monad.Instance.Delay.html#13159" class="Bound">f</a> <a id="13260" href="Categories.Object.Coproduct.html#532" class="Function Operator">,</a> <a id="13262" href="Categories.Object.Coproduct.html#512" class="Function">i₂</a> <a id="13265" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="13267" href="Categories.Object.Coproduct.html#532" class="Function Operator">[</a> <a id="13269" href="Monad.Instance.Delay.html#13159" class="Bound">f</a> <a id="13271" href="Categories.Object.Coproduct.html#532" class="Function Operator">,</a> <a id="13273" href="Monad.Instance.Delay.html#8028" class="Function">extend&#39;</a> <a id="13281" class="Symbol">(</a><a id="13282" href="Monad.Instance.Delay.html#2255" class="Function">▷</a> <a id="13284" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="13286" href="Monad.Instance.Delay.html#13159" class="Bound">f</a><a id="13287" class="Symbol">)</a> <a id="13289" href="Categories.Object.Coproduct.html#532" class="Function Operator">]</a> <a id="13291" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="13293" href="Monad.Instance.Delay.html#2116" class="Function">out</a> <a id="13297" href="Categories.Object.Coproduct.html#532" class="Function Operator">]</a> <a id="13299" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="13301" href="Monad.Instance.Delay.html#2116" class="Function">out</a>
<a id="13315" href="Monad.Instance.Delay.html#13203" class="Function">helper</a> <a id="13322" class="Symbol">=</a> <a id="13324" href="Categories.Morphism.Reasoning.Core.html#2048" class="Function">pullˡ</a> <a id="13330" href="Categories.Category.Cocartesian.html#2736" class="Function">∘[]</a> <a id="13334" href="Categories.Category.Core.html#3061" class="Function Operator">○</a> <a id="13336" class="Symbol">((</a><a id="13338" href="Categories.Object.Coproduct.html#816" class="Function">[]-cong₂</a> <a id="13347" href="Relation.Binary.Structures.html#1596" class="Function">refl</a> <a id="13352" class="Symbol">(</a><a id="13353" href="Monad.Instance.Delay.html#8294" class="Function">extendlaw</a> <a id="13363" class="Symbol">(</a><a id="13364" href="Monad.Instance.Delay.html#2255" class="Function">▷</a> <a id="13366" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="13368" href="Monad.Instance.Delay.html#13159" class="Bound">f</a><a id="13369" class="Symbol">)</a> <a id="13371" href="Categories.Category.Core.html#3061" class="Function Operator">○</a> <a id="13373" class="Symbol">(((</a><a id="13376" href="Categories.Object.Coproduct.html#816" class="Function">[]-cong₂</a> <a id="13385" class="Symbol">(</a><a id="13386" href="Categories.Morphism.Reasoning.Core.html#2048" class="Function">pullˡ</a> <a id="13392" href="Monad.Instance.Delay.html#2345" class="Function">laterlaw</a><a id="13400" class="Symbol">)</a> <a id="13402" href="Relation.Binary.Structures.html#1596" class="Function">refl</a><a id="13406" class="Symbol">)</a> <a id="13408" href="Categories.Category.Core.html#2837" class="Function Operator">⟩∘⟨refl</a><a id="13415" class="Symbol">)</a> <a id="13417" href="Categories.Category.Core.html#3061" class="Function Operator">○</a> <a id="13419" href="Relation.Binary.Structures.html#1622" class="Function">sym</a> <a id="13423" class="Symbol">(</a><a id="13424" href="Categories.Morphism.Reasoning.Core.html#2048" class="Function">pullˡ</a> <a id="13430" href="Categories.Category.Cocartesian.html#2736" class="Function">∘[]</a><a id="13433" class="Symbol">))))</a> <a id="13438" href="Categories.Category.Core.html#2837" class="Function Operator">⟩∘⟨refl</a><a id="13445" class="Symbol">)</a>
<a id="13457" href="Monad.Instance.Delay.html#13457" class="Function">helper₁</a> <a id="13465" class="Symbol">:</a> <a id="13467" href="Categories.Object.Coproduct.html#532" class="Function Operator">[</a> <a id="13469" href="Monad.Instance.Delay.html#13159" class="Bound">f</a> <a id="13471" href="Categories.Object.Coproduct.html#532" class="Function Operator">,</a> <a id="13473" href="Monad.Instance.Delay.html#8028" class="Function">extend&#39;</a> <a id="13481" class="Symbol">(</a><a id="13482" href="Monad.Instance.Delay.html#2255" class="Function">▷</a> <a id="13484" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="13486" href="Monad.Instance.Delay.html#13159" class="Bound">f</a><a id="13487" class="Symbol">)</a> <a id="13489" href="Categories.Object.Coproduct.html#532" class="Function Operator">]</a> <a id="13491" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="13493" href="Monad.Instance.Delay.html#2116" class="Function">out</a> <a id="13497" href="Categories.Category.Core.html#595" class="Function Operator">≈</a> <a id="13499" href="Monad.Instance.Delay.html#8028" class="Function">extend&#39;</a> <a id="13507" href="Monad.Instance.Delay.html#13159" class="Bound">f</a>
<a id="13519" href="Monad.Instance.Delay.html#13457" class="Function">helper₁</a> <a id="13527" class="Symbol">=</a> <a id="13529" href="Relation.Binary.Structures.html#1622" class="Function">sym</a> <a id="13533" class="Symbol">(</a><a id="13534" href="Monad.Instance.Delay.html#9642" class="Function">extend&#39;-unique</a> <a id="13549" href="Monad.Instance.Delay.html#13159" class="Bound">f</a> <a id="13551" class="Symbol">(</a><a id="13552" href="Categories.Object.Coproduct.html#532" class="Function Operator">[</a> <a id="13554" href="Monad.Instance.Delay.html#13159" class="Bound">f</a> <a id="13556" href="Categories.Object.Coproduct.html#532" class="Function Operator">,</a> <a id="13558" href="Monad.Instance.Delay.html#8028" class="Function">extend&#39;</a> <a id="13566" class="Symbol">(</a><a id="13567" href="Monad.Instance.Delay.html#2255" class="Function">▷</a> <a id="13569" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="13571" href="Monad.Instance.Delay.html#13159" class="Bound">f</a><a id="13572" class="Symbol">)</a> <a id="13574" href="Categories.Object.Coproduct.html#532" class="Function Operator">]</a> <a id="13576" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="13578" href="Monad.Instance.Delay.html#2116" class="Function">out</a><a id="13581" class="Symbol">)</a> <a id="13583" href="Monad.Instance.Delay.html#13203" class="Function">helper</a><a id="13589" class="Symbol">)</a>
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<a id="23125" class="Symbol">(</a><a id="23126" href="Monad.Instance.Delay.html#1644" class="Field">u</a> <a id="23128" class="Symbol">(</a><a id="23129" href="Categories.Object.Terminal.html#577" class="Function">!</a> <a id="23131" class="Symbol">(</a><a id="23132" href="Monad.Instance.Delay.html#1739" class="Field">coalgebras</a> <a id="23143" href="Monad.Instance.Delay.html#22532" class="Bound">Y</a><a id="23144" class="Symbol">)</a> <a id="23146" class="Symbol">{</a><a id="23147" class="Argument">A</a> <a id="23149" class="Symbol">=</a> <a id="23151" href="Monad.Instance.Delay.html#7879" class="Function">alg</a> <a id="23155" href="Monad.Instance.Delay.html#22540" class="Bound">g</a> <a id="23157" class="Symbol">})</a> <a id="23160" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="23162" href="Category.Ambient.html#2115" class="Function">idC</a><a id="23165" class="Symbol">)</a> <a id="23167" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="23169" href="Categories.Object.Coproduct.html#492" class="Function">i₁</a> <a id="23172" class="Symbol">{</a><a id="23173" class="Argument">B</a> <a id="23175" class="Symbol">=</a> <a id="23177" href="Monad.Instance.Delay.html#7773" class="Function">D₀</a> <a id="23180" href="Monad.Instance.Delay.html#22532" class="Bound">Y</a><a id="23181" class="Symbol">}</a> <a id="23183" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="23186" href="Categories.Category.Core.html#1145" class="Function">identityʳ</a> <a id="23196" href="Categories.Category.Core.html#2837" class="Function Operator">⟩∘⟨refl</a> <a id="23204" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
<a id="23216" href="Monad.Instance.Delay.html#8028" class="Function">extend&#39;</a> <a id="23224" href="Monad.Instance.Delay.html#22540" class="Bound">g</a> <a id="23274" href="Relation.Binary.Reasoning.Syntax.html#12283" class="Function Operator">∎</a>
<a id="DelayM.monad"></a><a id="23281" href="Monad.Instance.Delay.html#23281" class="Function">monad</a> <a id="23287" class="Symbol">:</a> <a id="23289" href="Categories.Monad.html#454" class="Record">Monad</a> <a id="23295" href="Category.Ambient.html#1757" class="Field">C</a>
<a id="23301" href="Monad.Instance.Delay.html#23281" class="Function">monad</a> <a id="23307" class="Symbol">=</a> <a id="23309" href="Categories.Monad.Construction.Kleisli.html#1000" class="Function">Kleisli⇒Monad</a> <a id="23323" href="Category.Ambient.html#1757" class="Field">C</a> <a id="23325" href="Monad.Instance.Delay.html#19476" class="Function">kleisli</a>
<a id="23338" class="Comment">-- redundant but convenient to have</a>
<a id="DelayM.functor"></a><a id="23378" href="Monad.Instance.Delay.html#23378" class="Function">functor</a> <a id="23386" class="Symbol">:</a> <a id="23388" href="Categories.Functor.html#283" class="Function">Endofunctor</a> <a id="23400" href="Category.Ambient.html#1757" class="Field">C</a>
<a id="23406" href="Monad.Instance.Delay.html#23378" class="Function">functor</a> <a id="23414" class="Symbol">=</a> <a id="23416" href="Categories.Monad.html#525" class="Field">Monad.F</a> <a id="23424" href="Monad.Instance.Delay.html#23281" class="Function">monad</a>
</pre>
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