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<h1 id="quotienting-the-delay-monad-by-weak-bisimilarity">Quotienting
the delay monad by weak bisimilarity</h1>
<p>The coinductive definition of the delay monad yields a ‘wrong’ kind
of equality called <em>strong bisimilarity</em>, which differentiates
between computations with different execution times, instead we want two
computations to be equal, when they both terminate with the same result
(or don’t terminate at all). This is called <em>weak bisimilarity</em>.
The ideas is then to quotient the delay monad with this ‘right’ kind of
equality and show that the result <span
class="math inline">\(\tilde{D}\)</span> is again a (strong) monad.</p>
<p>In category theory existence of <strong>coequalizers</strong>
corresponds to qoutienting, so we will express the quotiented delay
monad via the following coequalizer:</p>
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<h2 id="preliminaries">Preliminaries</h2>
<p>Existence of the coequalizer doesn’t suffice, we will need some
conditions having to do with preservation of the coequalizer, so let’s
first define what it means for a coequalizer to be preserved by an
endofunctor:</p>
<pre class="Agda"> <a id="preserves"></a><a id="2286" href="Monad.Instance.Delay.Quotienting.html#2286" class="Function">preserves</a> <a id="2296" class="Symbol">:</a> <a id="2298" class="Symbol">∀</a> <a id="2300" class="Symbol">{</a><a id="2301" href="Monad.Instance.Delay.Quotienting.html#2301" class="Bound">X</a> <a id="2303" href="Monad.Instance.Delay.Quotienting.html#2303" class="Bound">Y</a><a id="2304" class="Symbol">}</a> <a id="2306" class="Symbol">{</a><a id="2307" href="Monad.Instance.Delay.Quotienting.html#2307" class="Bound">f</a> <a id="2309" href="Monad.Instance.Delay.Quotienting.html#2309" class="Bound">g</a> <a id="2311" class="Symbol">:</a> <a id="2313" href="Monad.Instance.Delay.Quotienting.html#2301" class="Bound">X</a> <a id="2315" href="Categories.Category.Core.html#575" class="Function Operator">⇒</a> <a id="2317" href="Monad.Instance.Delay.Quotienting.html#2303" class="Bound">Y</a><a id="2318" class="Symbol">}</a> <a id="2320" class="Symbol">→</a> <a id="2322" href="Categories.Functor.html#283" class="Function">Endofunctor</a> <a id="2334" href="Category.Ambient.html#1757" class="Field">C</a> <a id="2336" class="Symbol">→</a> <a id="2338" href="Categories.Diagram.Coequalizer.html#1885" class="Record">Coequalizer</a> <a id="2350" href="Monad.Instance.Delay.Quotienting.html#2307" class="Bound">f</a> <a id="2352" href="Monad.Instance.Delay.Quotienting.html#2309" class="Bound">g</a> <a id="2354" class="Symbol">→</a> <a id="2356" href="Agda.Primitive.html#388" class="Primitive">Set</a> <a id="2360" class="Symbol">(</a><a id="2361" href="Monad.Instance.Delay.Quotienting.html#530" class="Bound">o</a> <a id="2363" href="Agda.Primitive.html#961" class="Primitive Operator">⊔</a> <a id="2365" href="Monad.Instance.Delay.Quotienting.html#532" class="Bound">ℓ</a> <a id="2367" href="Agda.Primitive.html#961" class="Primitive Operator">⊔</a> <a id="2369" href="Monad.Instance.Delay.Quotienting.html#534" class="Bound">e</a><a id="2370" class="Symbol">)</a>
<a id="2374" href="Monad.Instance.Delay.Quotienting.html#2286" class="Function">preserves</a> <a id="2384" class="Symbol">{</a><a id="2385" href="Monad.Instance.Delay.Quotienting.html#2385" class="Bound">X</a><a id="2386" class="Symbol">}</a> <a id="2388" class="Symbol">{</a><a id="2389" href="Monad.Instance.Delay.Quotienting.html#2389" class="Bound">Y</a><a id="2390" class="Symbol">}</a> <a id="2392" class="Symbol">{</a><a id="2393" href="Monad.Instance.Delay.Quotienting.html#2393" class="Bound">f</a><a id="2394" class="Symbol">}</a> <a id="2396" class="Symbol">{</a><a id="2397" href="Monad.Instance.Delay.Quotienting.html#2397" class="Bound">g</a><a id="2398" class="Symbol">}</a> <a id="2400" href="Monad.Instance.Delay.Quotienting.html#2400" class="Bound">F</a> <a id="2402" href="Monad.Instance.Delay.Quotienting.html#2402" class="Bound">coeq</a> <a id="2407" class="Symbol">=</a> <a id="2409" href="Categories.Diagram.Coequalizer.html#406" class="Record">IsCoequalizer</a> <a id="2423" class="Symbol">(</a><a id="2424" href="Categories.Functor.Core.html#806" class="Function">Functor.₁</a> <a id="2434" href="Monad.Instance.Delay.Quotienting.html#2400" class="Bound">F</a> <a id="2436" href="Monad.Instance.Delay.Quotienting.html#2393" class="Bound">f</a><a id="2437" class="Symbol">)</a> <a id="2439" class="Symbol">(</a><a id="2440" href="Categories.Functor.Core.html#806" class="Function">Functor.₁</a> <a id="2450" href="Monad.Instance.Delay.Quotienting.html#2400" class="Bound">F</a> <a id="2452" href="Monad.Instance.Delay.Quotienting.html#2397" class="Bound">g</a><a id="2453" class="Symbol">)</a> <a id="2455" class="Symbol">(</a><a id="2456" href="Categories.Functor.Core.html#806" class="Function">Functor.₁</a> <a id="2466" href="Monad.Instance.Delay.Quotienting.html#2400" class="Bound">F</a> <a id="2468" class="Symbol">(</a><a id="2469" href="Categories.Diagram.Coequalizer.html#1963" class="Field">Coequalizer.arr</a> <a id="2485" href="Monad.Instance.Delay.Quotienting.html#2402" class="Bound">coeq</a><a id="2489" class="Symbol">))</a>
</pre>
<p>We will now show that the following conditions are equivalent:</p>
<ol type="1">
<li>For every <span class="math inline">\(X\)</span>, the coequalizer is
preserved by <span class="math inline">\(D\)</span></li>
<li>every <span class="math inline">\(\tilde{D}X\)</span> extends to a
search-algebra, so that each <span class="math inline">\(ρ_X\)</span> is
a D-algebra morphism</li>
<li>for every <span class="math inline">\(X\)</span>, <span
class="math inline">\((\tilde{D}X, ρ ∘ now : X → \tilde{D}X)\)</span> is
a stable free elgot algebra on X, <span
class="math inline">\(ρ_X\)</span> is a D-algebra morphism and <span
class="math inline">\(ρ_X = ((ρ_X ∘ now + id) ∘ out)^\#\)</span></li>
<li><span class="math inline">\(\tilde{D}\)</span> extends to a strong
monad, so that <span class="math inline">\(ρ\)</span> is a strong monad
morphism</li>
</ol>
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<a id="2999" class="Keyword">open</a> <a id="3004" href="Monad.Instance.Delay.html#1692" class="Module">DelayM</a> <a id="3011" href="Monad.Instance.Delay.Quotienting.html#2977" class="Bound">D</a> <a id="3013" class="Keyword">renaming</a> <a id="3022" class="Symbol">(</a><a id="3023" href="Monad.Instance.Delay.html#23378" class="Function">functor</a> <a id="3031" class="Symbol">to</a> <a id="3034" class="Function">D-Functor</a><a id="3043" class="Symbol">;</a> <a id="3045" href="Monad.Instance.Delay.html#23281" class="Function">monad</a> <a id="3051" class="Symbol">to</a> <a id="3054" class="Function">D-Monad</a><a id="3061" class="Symbol">;</a> <a id="3063" href="Monad.Instance.Delay.html#19476" class="Function">kleisli</a> <a id="3071" class="Symbol">to</a> <a id="3074" class="Function">D-Kleisli</a><a id="3083" class="Symbol">)</a>
<a id="3090" class="Keyword">open</a> <a id="3095" href="Categories.Functor.Core.html#248" class="Module">Functor</a> <a id="3103" href="Monad.Instance.Delay.Quotienting.html#3034" class="Function">D-Functor</a> <a id="3113" class="Keyword">using</a> <a id="3119" class="Symbol">()</a> <a id="3122" class="Keyword">renaming</a> <a id="3131" class="Symbol">(</a><a id="3132" href="Categories.Functor.Core.html#455" class="Field">F₁</a> <a id="3135" class="Symbol">to</a> <a id="3138" class="Field">D₁</a><a id="3140" class="Symbol">;</a> <a id="3142" href="Categories.Functor.Core.html#565" class="Field">homomorphism</a> <a id="3155" class="Symbol">to</a> <a id="3158" class="Field">D-homomorphism</a><a id="3172" class="Symbol">;</a> <a id="3174" href="Categories.Functor.Core.html#696" class="Field">F-resp-≈</a> <a id="3183" class="Symbol">to</a> <a id="3186" class="Field">D-resp-≈</a><a id="3194" class="Symbol">;</a> <a id="3196" href="Categories.Functor.Core.html#511" class="Field">identity</a> <a id="3205" class="Symbol">to</a> <a id="3208" class="Field">D-identity</a><a id="3218" class="Symbol">)</a>
<a id="3224" class="Keyword">open</a> <a id="3229" href="Monad.Instance.Delay.Quotienting.html#245" class="Module">RMonad</a> <a id="3236" href="Monad.Instance.Delay.Quotienting.html#3074" class="Function">D-Kleisli</a> <a id="3246" class="Keyword">using</a> <a id="3252" class="Symbol">(</a><a id="3253" href="Categories.Monad.Relative.html#895" class="Field">extend</a><a id="3259" class="Symbol">;</a> <a id="3261" href="Categories.Monad.Relative.html#1306" class="Field">extend-≈</a><a id="3269" class="Symbol">)</a> <a id="3271" class="Keyword">renaming</a> <a id="3280" class="Symbol">(</a><a id="3281" href="Categories.Monad.Relative.html#1078" class="Field">assoc</a> <a id="3287" class="Symbol">to</a> <a id="3290" class="Field">k-assoc</a><a id="3297" class="Symbol">;</a> <a id="3299" href="Categories.Monad.Relative.html#953" class="Field">identityʳ</a> <a id="3309" class="Symbol">to</a> <a id="3312" class="Field">k-identityʳ</a><a id="3323" class="Symbol">;</a> <a id="3325" href="Categories.Monad.Relative.html#1025" class="Field">identityˡ</a> <a id="3335" class="Symbol">to</a> <a id="3338" class="Field">k-identityˡ</a><a id="3349" class="Symbol">)</a>
<a id="3355" class="Keyword">open</a> <a id="3360" href="Categories.Monad.html#454" class="Module">Monad</a> <a id="3366" href="Monad.Instance.Delay.Quotienting.html#3054" class="Function">D-Monad</a> <a id="3374" class="Keyword">using</a> <a id="3380" class="Symbol">()</a> <a id="3383" class="Keyword">renaming</a> <a id="3392" class="Symbol">(</a><a id="3393" href="Categories.Monad.html#759" class="Field">assoc</a> <a id="3399" class="Symbol">to</a> <a id="3402" class="Field">M-assoc</a><a id="3409" class="Symbol">;</a> <a id="3411" href="Categories.Monad.html#953" class="Field">identityʳ</a> <a id="3421" class="Symbol">to</a> <a id="3424" class="Field">M-identityʳ</a><a id="3435" class="Symbol">)</a>
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<a id="3463" class="Keyword">open</a> <a id="3468" href="Category.Ambient.html#9069" class="Module">M</a> <a id="3470" href="Category.Ambient.html#1757" class="Field">C</a>
<a id="3476" class="Keyword">open</a> <a id="3481" href="Category.Ambient.html#9087" class="Module">MR</a> <a id="3484" href="Category.Ambient.html#1757" class="Field">C</a>
<a id="3490" class="Keyword">open</a> <a id="3495" href="Categories.Category.Core.html#1530" class="Module">Equiv</a>
<a id="3506" class="Keyword">module</a> <a id="3513" href="Monad.Instance.Delay.Quotienting.html#3513" class="Module">Quotiented</a> <a id="3524" class="Symbol">(</a><a id="3525" href="Monad.Instance.Delay.Quotienting.html#3525" class="Bound">coeqs</a> <a id="3531" class="Symbol">:</a> <a id="3533" class="Symbol">∀</a> <a id="3535" href="Monad.Instance.Delay.Quotienting.html#3535" class="Bound">X</a> <a id="3537" class="Symbol">→</a> <a id="3539" href="Categories.Diagram.Coequalizer.html#1885" class="Record">Coequalizer</a> <a id="3551" class="Symbol">(</a><a id="3552" href="Categories.Monad.Relative.html#895" class="Function">extend</a> <a id="3559" class="Symbol">(</a><a id="3560" href="Monad.Instance.Delay.html#5371" class="Function">ι</a> <a id="3562" class="Symbol">{</a><a id="3563" href="Monad.Instance.Delay.Quotienting.html#3535" class="Bound">X</a><a id="3564" class="Symbol">}))</a> <a id="3568" class="Symbol">(</a><a id="3569" href="Monad.Instance.Delay.Quotienting.html#3138" class="Function">D₁</a> <a id="3572" href="Categories.Object.Product.Core.html#526" class="Function">π₁</a><a id="3574" class="Symbol">))</a> <a id="3577" class="Keyword">where</a>
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<a id="9268" class="Symbol">(</a><a id="9269" href="Category.Ambient.html#2115" class="Function">idC</a> <a id="9273" href="Categories.Category.Cocartesian.html#2133" class="Function Operator">+₁</a> <a id="9276" href="Monad.Instance.Delay.html#5371" class="Function">ι</a><a id="9277" class="Symbol">)</a> <a id="9279" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="9281" href="Categories.Morphism.html#2006" class="Field">_≅_.from</a> <a id="9290" href="Monad.Instance.Delay.html#4993" class="Function">nno-iso</a> <a id="9298" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="9300" href="Categories.Object.Coproduct.html#532" class="Function Operator">[</a> <a id="9302" href="Categories.Category.BinaryProducts.html#1396" class="Function Operator">⟨</a> <a id="9304" href="Monad.Instance.Delay.Quotienting.html#7323" class="Bound">f</a> <a id="9306" href="Categories.Category.BinaryProducts.html#1396" class="Function Operator">,</a> <a id="9308" href="Categories.Object.NaturalNumbers.Parametrized.html#960" class="Function">z</a> <a id="9310" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="9312" href="Categories.Object.Terminal.html#577" class="Function">Terminal.!</a> <a id="9323" href="Categories.Category.Cartesian.html#777" class="Function">terminal</a> <a id="9332" href="Categories.Category.BinaryProducts.html#1396" class="Function Operator">⟩</a> <a id="9334" href="Categories.Object.Coproduct.html#532" class="Function Operator">,</a> <a id="9336" class="Symbol">(</a><a id="9337" href="Monad.Instance.Delay.Quotienting.html#7323" class="Bound">f</a> <a id="9339" href="Categories.Category.BinaryProducts.html#1465" class="Function Operator">⁂</a> <a id="9341" href="Categories.Object.NaturalNumbers.Parametrized.html#974" class="Function">s</a><a id="9342" class="Symbol">)</a> <a id="9344" href="Categories.Object.Coproduct.html#532" class="Function Operator">]</a> <a id="9346" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="9348" href="Categories.Morphism.html#2006" class="Field">_≅_.from</a> <a id="9357" href="Monad.Instance.Delay.html#4993" class="Function">nno-iso</a> <a id="9365" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="9368" class="Symbol">(</a><a id="9369" href="Categories.Category.Core.html#2734" class="Function Operator">refl⟩∘⟨</a> <a id="9377" href="Categories.Category.Core.html#2734" class="Function Operator">refl⟩∘⟨</a> <a id="9385" href="Categories.Object.Coproduct.html#816" class="Function">[]-cong₂</a> <a id="9394" class="Symbol">(</a><a id="9395" href="Categories.Object.Product.Core.html#854" class="Function">⟨⟩-cong₂</a> <a id="9404" href="Relation.Binary.Structures.html#1596" class="Function">refl</a> <a id="9409" class="Symbol">(</a><a id="9410" href="Categories.Category.Core.html#2734" class="Function Operator">refl⟩∘⟨</a> <a id="9418" href="Categories.Object.Terminal.html#605" class="Function">Terminal.!-unique</a> <a id="9436" href="Categories.Category.Cartesian.html#777" class="Function">terminal</a> <a id="9445" class="Symbol">(</a><a id="9446" href="Categories.Object.Terminal.html#577" class="Function">Terminal.!</a> <a id="9457" href="Categories.Category.Cartesian.html#777" class="Function">terminal</a> <a id="9466" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="9468" href="Monad.Instance.Delay.Quotienting.html#7323" class="Bound">f</a><a id="9469" class="Symbol">)))</a> <a id="9473" href="Relation.Binary.Structures.html#1596" class="Function">refl</a> <a id="9478" href="Categories.Category.Core.html#2837" class="Function Operator">⟩∘⟨refl</a><a id="9485" class="Symbol">)</a> <a id="9487" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
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<a id="10471" href="Monad.Instance.Delay.Quotienting.html#3651" class="Function">ρ</a> <a id="10473" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="10475" href="Categories.Monad.Relative.html#895" class="Function">extend</a> <a id="10482" class="Symbol">(</a><a id="10483" href="Categories.Monad.Relative.html#895" class="Function">extend</a> <a id="10490" href="Monad.Instance.Delay.html#5371" class="Function">ι</a> <a id="10492" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="10494" href="Monad.Instance.Delay.html#2135" class="Function">now</a> <a id="10498" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="10500" class="Symbol">(</a><a id="10501" href="Monad.Instance.Delay.Quotienting.html#7323" class="Bound">f</a> <a id="10503" href="Categories.Category.BinaryProducts.html#1465" class="Function Operator">⁂</a> <a id="10505" href="Category.Ambient.html#2115" class="Function">idC</a><a id="10508" class="Symbol">))</a> <a id="10511" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="10514" href="Categories.Morphism.Reasoning.Core.html#2347" class="Function">pushʳ</a> <a id="10520" href="Monad.Instance.Delay.Quotienting.html#3290" class="Function">k-assoc</a> <a id="10528" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
<a id="10542" class="Symbol">(</a><a id="10543" href="Monad.Instance.Delay.Quotienting.html#3651" class="Function">ρ</a> <a id="10545" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="10547" href="Categories.Monad.Relative.html#895" class="Function">extend</a> <a id="10554" href="Monad.Instance.Delay.html#5371" class="Function">ι</a><a id="10555" class="Symbol">)</a> <a id="10557" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="10559" href="Monad.Instance.Delay.Quotienting.html#3138" class="Function">D₁</a> <a id="10562" class="Symbol">(</a><a id="10563" href="Monad.Instance.Delay.Quotienting.html#7323" class="Bound">f</a> <a id="10565" href="Categories.Category.BinaryProducts.html#1465" class="Function Operator">⁂</a> <a id="10567" href="Category.Ambient.html#2115" class="Function">idC</a><a id="10570" class="Symbol">)</a> <a id="10572" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="10575" href="Categories.Morphism.Reasoning.Core.html#2485" class="Function">pushˡ</a> <a id="10581" class="Symbol">(</a><a id="10582" href="Categories.Diagram.Coequalizer.html#488" class="Function">Coequalizer.equality</a> <a id="10603" class="Symbol">(</a><a id="10604" href="Monad.Instance.Delay.Quotienting.html#3525" class="Bound">coeqs</a> <a id="10610" href="Monad.Instance.Delay.Quotienting.html#7320" class="Bound">Y</a><a id="10611" class="Symbol">))</a> <a id="10614" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
<a id="10628" href="Monad.Instance.Delay.Quotienting.html#3651" class="Function">ρ</a> <a id="10630" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="10632" href="Monad.Instance.Delay.Quotienting.html#3138" class="Function">D₁</a> <a id="10635" href="Categories.Object.Product.Core.html#526" class="Function">π₁</a> <a id="10638" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="10640" href="Monad.Instance.Delay.Quotienting.html#3138" class="Function">D₁</a> <a id="10643" class="Symbol">(</a><a id="10644" href="Monad.Instance.Delay.Quotienting.html#7323" class="Bound">f</a> <a id="10646" href="Categories.Category.BinaryProducts.html#1465" class="Function Operator">⁂</a> <a id="10648" href="Category.Ambient.html#2115" class="Function">idC</a><a id="10651" class="Symbol">)</a> <a id="10653" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="10656" href="Categories.Category.Core.html#2734" class="Function Operator">refl⟩∘⟨</a> <a id="10664" href="Relation.Binary.Structures.html#1622" class="Function">sym</a> <a id="10668" href="Monad.Instance.Delay.Quotienting.html#3158" class="Function">D-homomorphism</a> <a id="10683" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
<a id="10697" href="Monad.Instance.Delay.Quotienting.html#3651" class="Function">ρ</a> <a id="10699" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="10701" href="Monad.Instance.Delay.Quotienting.html#3138" class="Function">D₁</a> <a id="10704" class="Symbol">(</a><a id="10705" href="Categories.Object.Product.Core.html#526" class="Function">π₁</a> <a id="10708" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="10710" class="Symbol">(</a><a id="10711" href="Monad.Instance.Delay.Quotienting.html#7323" class="Bound">f</a> <a id="10713" href="Categories.Category.BinaryProducts.html#1465" class="Function Operator">⁂</a> <a id="10715" href="Category.Ambient.html#2115" class="Function">idC</a><a id="10718" class="Symbol">))</a> <a id="10721" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="10724" class="Symbol">(</a><a id="10725" href="Categories.Category.Core.html#2734" class="Function Operator">refl⟩∘⟨</a> <a id="10733" class="Symbol">(</a><a id="10734" href="Monad.Instance.Delay.Quotienting.html#3186" class="Function">D-resp-≈</a> <a id="10743" href="Categories.Object.Product.Core.html#603" class="Function">project₁</a><a id="10751" class="Symbol">))</a> <a id="10754" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
<a id="10768" href="Monad.Instance.Delay.Quotienting.html#3651" class="Function">ρ</a> <a id="10770" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="10772" href="Monad.Instance.Delay.Quotienting.html#3138" class="Function">D₁</a> <a id="10775" class="Symbol">(</a><a id="10776" href="Monad.Instance.Delay.Quotienting.html#7323" class="Bound">f</a> <a id="10778" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="10780" href="Categories.Object.Product.Core.html#526" class="Function">π₁</a><a id="10782" class="Symbol">)</a> <a id="10784" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="10787" href="Categories.Morphism.Reasoning.Core.html#2347" class="Function">pushʳ</a> <a id="10793" href="Monad.Instance.Delay.Quotienting.html#3158" class="Function">D-homomorphism</a> <a id="10808" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
<a id="10822" class="Symbol">(</a><a id="10823" href="Monad.Instance.Delay.Quotienting.html#3651" class="Function">ρ</a> <a id="10825" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="10827" href="Monad.Instance.Delay.Quotienting.html#3138" class="Function">D₁</a> <a id="10830" href="Monad.Instance.Delay.Quotienting.html#7323" class="Bound">f</a><a id="10831" class="Symbol">)</a> <a id="10833" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="10835" href="Monad.Instance.Delay.Quotienting.html#3138" class="Function">D₁</a> <a id="10838" href="Categories.Object.Product.Core.html#526" class="Function">π₁</a> <a id="10841" href="Relation.Binary.Reasoning.Syntax.html#12283" class="Function Operator">∎</a><a id="10842" class="Symbol">)</a>
<a id="10856" class="Keyword">where</a>
<a id="10877" class="Keyword">open</a> <a id="10882" href="Categories.Diagram.Coequalizer.html#1885" class="Module">Coequalizer</a> <a id="10894" class="Symbol">(</a><a id="10895" href="Monad.Instance.Delay.Quotienting.html#3525" class="Bound">coeqs</a> <a id="10901" href="Monad.Instance.Delay.Quotienting.html#7316" class="Bound">X</a><a id="10902" class="Symbol">)</a> <a id="10904" class="Keyword">using</a> <a id="10910" class="Symbol">(</a><a id="10911" href="Categories.Diagram.Coequalizer.html#523" class="Function">coequalize</a><a id="10921" class="Symbol">;</a> <a id="10923" href="Categories.Diagram.Coequalizer.html#488" class="Function">equality</a><a id="10931" class="Symbol">)</a> <a id="10933" class="Keyword">renaming</a> <a id="10942" class="Symbol">(</a><a id="10943" href="Categories.Diagram.Coequalizer.html#576" class="Function">universal</a> <a id="10953" class="Symbol">to</a> <a id="10956" class="Function">coeq-universal</a><a id="10970" class="Symbol">)</a>
<a id="10979" href="Monad.Instance.Delay.Quotienting.html#10979" class="Function">ρ-natural</a> <a id="10989" class="Symbol">:</a> <a id="10991" href="Categories.NaturalTransformation.Core.html#466" class="Record">NaturalTransformation</a> <a id="11013" href="Monad.Instance.Delay.Quotienting.html#3034" class="Function">D-Functor</a> <a id="11023" href="Monad.Instance.Delay.Quotienting.html#6551" class="Function">Ď-Functor</a>
<a id="11039" href="Monad.Instance.Delay.Quotienting.html#10979" class="Function">ρ-natural</a> <a id="11049" class="Symbol">=</a> <a id="11051" href="Categories.NaturalTransformation.Core.html#1750" class="Function">ntHelper</a> <a id="11060" class="Symbol">(</a><a id="11061" class="Keyword">record</a>
<a id="11077" class="Symbol">{</a> <a id="11079" href="Categories.NaturalTransformation.Core.html#1637" class="Field">η</a> <a id="11081" class="Symbol">=</a> <a id="11083" class="Symbol">λ</a> <a id="11085" href="Monad.Instance.Delay.Quotienting.html#11085" class="Bound">X</a> <a id="11087" class="Symbol">→</a> <a id="11089" href="Monad.Instance.Delay.Quotienting.html#3651" class="Function">ρ</a> <a id="11091" class="Symbol">{</a><a id="11092" href="Monad.Instance.Delay.Quotienting.html#11085" class="Bound">X</a><a id="11093" class="Symbol">}</a>
<a id="11104" class="Symbol">;</a> <a id="11106" href="Categories.NaturalTransformation.Core.html#1681" class="Field">commute</a> <a id="11114" class="Symbol">=</a> <a id="11116" class="Symbol">λ</a> <a id="11118" class="Symbol">{</a><a id="11119" href="Monad.Instance.Delay.Quotienting.html#11119" class="Bound">X</a><a id="11120" class="Symbol">}</a> <a id="11122" class="Symbol">{</a><a id="11123" href="Monad.Instance.Delay.Quotienting.html#11123" class="Bound">Y</a><a id="11124" class="Symbol">}</a> <a id="11126" href="Monad.Instance.Delay.Quotienting.html#11126" class="Bound">f</a> <a id="11128" class="Symbol">→</a> <a id="11130" href="Categories.Diagram.Coequalizer.html#576" class="Function">Coequalizer.universal</a> <a id="11152" class="Symbol">(</a><a id="11153" href="Monad.Instance.Delay.Quotienting.html#3525" class="Bound">coeqs</a> <a id="11159" href="Monad.Instance.Delay.Quotienting.html#11119" class="Bound">X</a><a id="11160" class="Symbol">)</a>
<a id="11170" class="Symbol">})</a>
</pre>
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