Finish merge and split symmetric monoidal functors

1 files changed, 0 insertions, 214 deletions

diff --git a/Data/WiringDiagram/Looped/Monoidal.agda b/Data/WiringDiagram/Looped/Monoidal.agda
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--- a/Data/WiringDiagram/Looped/Monoidal.agda
+++ /dev/null
@@ -1,214 +0,0 @@
-{-# OPTIONS --without-K --safe #-}
-{-# OPTIONS --lossy-unification #-}
-
-open import Categories.Category using (Category)
-open import Categories.Category.Monoidal.Bundle using (MonoidalCategory)
-open import Categories.Functor using (Functor; _∘F_)
-open import Categories.Functor.Monoidal using (StrongMonoidalFunctor; MonoidalFunctor; IsMonoidalFunctor)
-open import Category.Dagger.2-Poset using (Map)
-open import Category.Dagger.Semiadditive using (IdempotentSemiadditiveDagger)
-open import Category.KaroubiComplete using (KaroubiComplete)
-open import Data.WiringDiagram.Monoidal using (BWD-MC)
-open import Level using (Level; suc; _⊔_)
-
-open MonoidalCategory using (U)
-
-module Data.WiringDiagram.Looped.Monoidal
-    {o ℓ e o′ ℓ′ e′ : Level}
-    {𝒞 : Category o ℓ e}
-    {𝒟 : MonoidalCategory o′ ℓ′ e′}
-    {S : IdempotentSemiadditiveDagger 𝒞}
-    (let module S = IdempotentSemiadditiveDagger S)
-    (let S′ = S.semiadditiveDagger)
-    (karoubiComplete : KaroubiComplete (U 𝒟))
-    (F : MonoidalFunctor (BWD-MC S′) 𝒟)
-  where
-
-module F = MonoidalFunctor F
-
-import Categories.Category.Monoidal.Reasoning as ⊗-Reasoning
-import Categories.Morphism.Reasoning as ⇒-Reasoning
-
-open import Categories.Category.Product using (_⁂_)
-open import Categories.Functor.Properties using ([_]-resp-square)
-open import Categories.NaturalTransformation using (NaturalTransformation; ntHelper)
-open import Data.Product using (_,_)
-open import Data.WiringDiagram.Balanced S′ using (Include; Push; Pull)
-open import Data.WiringDiagram.Core S′ using (loop)
-open import Data.WiringDiagram.Equalities S using (loop∘loop; loop∘push∘loop; loop∘pull∘loop)
-open import Data.WiringDiagram.Looped.Core {S = S} karoubiComplete F.F using (Merge; Looped; π; forget; L; π∘l; forget∘π; π∘forget; l∘forget; l∘l)
-open import Data.WiringDiagram.Monoidal S′ using (Push-MF; loop⊞loop; module BalancedPush)
-
-module BWD = BWD-MC S′
-module Merge = Functor Merge
-module Push = Functor Push
-module Push-MF = StrongMonoidalFunctor Push-MF
-module maps-MC = MonoidalCategory S.maps-MC
-module S-MC = MonoidalCategory S.monoidalCategory
-module 𝒞 = Category 𝒞
-module 𝒟 = MonoidalCategory 𝒟
-
-open BWD using () renaming (_∘_ to _∘′_; _⊗₁_ to _⊞₁_)
-open BalancedPush using (Push-⊞₁; Push-assoc; Push-π₂; Push-π₁)
-open Map using (map; entire)
-open maps-MC using () renaming (_⊗₁_ to _⊗₁′_)
-open 𝒟 using (_⇒_; _∘_; id; _≈_; _⊗₀_; _⊗₁_)
-open S using (_⊕_; _×₁_)
-
-ε : 𝒟.unit ⇒ Looped maps-MC.unit
-ε = π maps-MC.unit ∘ F.ε
-
-η : (X Y : 𝒞.Obj) → Looped X ⊗₀ Looped Y ⇒ Looped (X maps-MC.⊗₀ Y)
-η X Y = π (X maps-MC.⊗₀ Y) ∘ F.⊗-homo.η (X , Y) ∘ forget X ⊗₁ forget Y
-
-private module Shorthands where
-
-  φ : {X Y : 𝒞.Obj} → F.₀ X ⊗₀ F.₀ Y ⇒ F.₀ (X maps-MC.⊗₀ Y)
-  φ {X} {Y} = F.⊗-homo.η (X , Y)
-
-  fo : {X : 𝒞.Obj} → Looped X ⇒ F.₀ X
-  fo {X} = forget X
-
-  π′ : {X : 𝒞.Obj} → F.₀ X ⇒ Looped X
-  π′ {X} = π X
-
-  L′ : {X : 𝒞.Obj} → F.₀ X ⇒ F.₀ X
-  L′ {X} = L X
-
-comm
-    : {X X′ Y Y′ : 𝒞.Obj}
-      (f : X maps-MC.⇒ X′)
-      (g : Y maps-MC.⇒ Y′)
-    → η X′ Y′ ∘ Merge.₁ f ⊗₁ Merge.₁ g 𝒟.≈ Merge.₁ (f maps-MC.⊗₁ g) ∘ η X Y
-comm {X} {X′} {Y} {Y′} f g = begin
-    (π′ ∘ φ ∘ fo ⊗₁ fo) ∘ (π′ ∘ F.₁ (Push.₁ f′) ∘ fo) ⊗₁ (π′ ∘ F.₁ (Push.₁ g′) ∘ fo)  ≈⟨ pullʳ (pullʳ (sym ⊗-distrib-over-∘)) ⟩
-    π′ ∘ φ ∘ (fo ∘ π X′ ∘ F.₁ (Push.₁ f′) ∘ fo) ⊗₁ (fo ∘ π Y′ ∘ F.₁ (Push.₁ g′) ∘ fo) ≈⟨ refl⟩∘⟨ refl⟩∘⟨ pullˡ (forget∘π X′) ⟩⊗⟨ pullˡ (forget∘π Y′) ⟩
-    π′ ∘ φ ∘ (L X′ ∘ F.₁ (Push.₁ f′) ∘ fo) ⊗₁ (L Y′ ∘ F.₁ (Push.₁ g′) ∘ fo)           ≈⟨ refl⟩∘⟨ refl⟩∘⟨ pushˡ F.homomorphism ⟩⊗⟨ pushˡ F.homomorphism  ⟨
-    π′ ∘ φ ∘ (F.₁ (loop ∘′ Push.₁ f′) ∘ fo) ⊗₁ (F.₁ (loop ∘′ Push.₁ g′) ∘ fo)         ≈⟨ refl⟩∘⟨ refl⟩∘⟨ ⊗-distrib-over-∘ ⟩
-    π′ ∘ φ ∘ F.₁ (loop ∘′ Push.₁ f′) ⊗₁ F.₁ (loop ∘′ Push.₁ g′) ∘ fo ⊗₁ fo            ≈⟨ refl⟩∘⟨ extendʳ (F.⊗-homo.commute _) ⟩  
-    π′ ∘ F.₁ ((loop ∘′ Push.₁ f′) ⊞₁ (loop ∘′ Push.₁ g′)) ∘ φ ∘ fo ⊗₁ fo              ≈⟨ refl⟩∘⟨ F.F-resp-≈ (⊗-Reasoning.⊗-distrib-over-∘ BWD.monoidal) ⟩∘⟨refl ⟩
-    π′ ∘ F.₁ (loop ⊞₁ loop ∘′ Push.₁ f′ ⊞₁ Push.₁ g′) ∘ φ ∘ fo ⊗₁ fo                  ≈⟨ refl⟩∘⟨ F.F-resp-≈ (BWD.∘-resp-≈ˡ loop⊞loop) ⟩∘⟨refl ⟩
-    π′ ∘ F.₁ (loop ∘′ Push.₁ f′ ⊞₁ Push.₁ g′) ∘ φ ∘ fo ⊗₁ fo                          ≈⟨ refl⟩∘⟨ F.F-resp-≈ (BWD.∘-resp-≈ʳ (Push-⊞₁ f′ g′)) ⟩∘⟨refl ⟩
-    π′ ∘ F.₁ (loop ∘′ Push.₁ (f′ ×₁ g′)) ∘ φ ∘ fo ⊗₁ fo                               ≈⟨ refl⟩∘⟨ F.F-resp-≈ (loop∘push∘loop (f′ ×₁ g′) (entire (f ⊗₁′ g))) ⟩∘⟨refl ⟨
-    π′ ∘ F.₁ (loop ∘′ Push.₁ (f′ ×₁ g′) ∘′ loop) ∘ φ ∘ fo ⊗₁ fo                       ≈⟨ refl⟩∘⟨ pushˡ F.homomorphism ⟩
-    π′ ∘ L (X′ ⊕ Y′) ∘ F.₁ (Push.₁ (f′ ×₁ g′) ∘′ loop) ∘ φ ∘ fo ⊗₁ fo                 ≈⟨ refl⟩∘⟨ refl⟩∘⟨ pushˡ F.homomorphism ⟩
-    π′ ∘ L (X′ ⊕ Y′) ∘ F.₁ (Push.₁ (f′ ×₁ g′)) ∘ L (X ⊕ Y) ∘ φ ∘ fo ⊗₁ fo             ≈⟨ pullˡ (π∘l (X′ ⊕ Y′)) ⟩
-    π′ ∘ F.₁ (Push.₁ (f′ ×₁ g′)) ∘ L (X ⊕ Y) ∘ φ ∘ fo ⊗₁ fo                           ≈⟨ pushʳ (pushʳ (pushˡ (sym (forget∘π (X ⊕ Y))))) ⟩
-    (π′ ∘ F.₁ (Push.₁ (f′ ×₁ g′)) ∘ forget (X ⊕ Y)) ∘ π (X ⊕ Y) ∘ φ ∘ fo ⊗₁ fo        ∎
-  where
-    f′ : X 𝒞.⇒ X′
-    f′ = map f
-    g′ : Y 𝒞.⇒ Y′
-    g′ = map g
-    open Shorthands
-    open 𝒟.Equiv
-    open ⊗-Reasoning 𝒟.monoidal
-    open ⇒-Reasoning (U 𝒟)
-
-⊗-homo : NaturalTransformation (𝒟.⊗ ∘F (Merge ⁂ Merge)) (Merge ∘F maps-MC.⊗)
-⊗-homo = ntHelper record
-    { η = λ (X , Y) → η X Y
-    ; commute = λ (f , g) → comm f g
-    }
-
-associativity
-    : {X Y Z : 𝒞.Obj}
-    → Merge.₁ maps-MC.associator.from ∘ η (X ⊕ Y) Z ∘ η X Y ⊗₁ id ≈ η X (Y ⊕ Z) ∘ id ⊗₁ η Y Z ∘ 𝒟.associator.from
-associativity {X} {Y} {Z} = begin
-    (π′ ∘ F.₁ (Push.₁ S.assocˡ) ∘ fo) ∘ η (X ⊕ Y) Z ∘ η X Y ⊗₁ id           ≈⟨ pullʳ (pullʳ (extendʳ (pullˡ (forget∘π ((X ⊕ Y) ⊕ Z))))) ⟩
-    π′ ∘ F.₁ (Push.₁ S.assocˡ) ∘ L′ ∘ (φ ∘ fo ⊗₁ fo) ∘ η X Y ⊗₁ id          ≈⟨ refl⟩∘⟨ refl⟩∘⟨ refl⟩∘⟨ pullʳ merge₁ʳ ⟩
-    π′ ∘ F.₁ (Push.₁ S.assocˡ) ∘ L′ ∘ φ ∘ (fo ∘ π′ ∘ φ ∘ fo ⊗₁ fo) ⊗₁ fo    ≈⟨ refl⟩∘⟨ refl⟩∘⟨ refl⟩∘⟨ refl⟩∘⟨ pullˡ (forget∘π (X ⊕ Y)) ⟩⊗⟨refl ⟩
-    π′ ∘ F.₁ (Push.₁ S.assocˡ) ∘ L′ ∘ φ ∘ (L′ ∘ φ ∘ fo ⊗₁ fo) ⊗₁ fo         ≈⟨ refl⟩∘⟨ refl⟩∘⟨ refl⟩∘⟨ refl⟩∘⟨ refl⟩⊗⟨ (l∘forget Z) ⟨
-    π′ ∘ F.₁ (Push.₁ S.assocˡ) ∘ L′ ∘ φ ∘ (L′ ∘ φ ∘ fo ⊗₁ fo) ⊗₁ (L′ ∘ fo)  ≈⟨ refl⟩∘⟨ refl⟩∘⟨ refl⟩∘⟨ refl⟩∘⟨ ⊗-distrib-over-∘ ⟩
-    π′ ∘ F.₁ (Push.₁ S.assocˡ) ∘ L′ ∘ φ ∘ L′ ⊗₁ L′ ∘ (φ ∘ fo ⊗₁ fo) ⊗₁ fo   ≈⟨ refl⟩∘⟨ refl⟩∘⟨ refl⟩∘⟨ extendʳ (F.⊗-homo.commute _) ⟩
-    π′ ∘ F.₁ (Push.₁ S.assocˡ) ∘ L′ ∘ F.₁ (loop ⊞₁ loop) ∘ φ ∘ _ ⊗₁ fo      ≈⟨ refl⟩∘⟨ refl⟩∘⟨ refl⟩∘⟨ F.F-resp-≈ loop⊞loop ⟩∘⟨refl ⟩
-    π′ ∘ F.₁ (Push.₁ S.assocˡ) ∘ L′ ∘ L′ ∘ φ ∘ (φ ∘ fo ⊗₁ fo) ⊗₁ fo         ≈⟨ refl⟩∘⟨ refl⟩∘⟨ pullˡ (l∘l ((X ⊕ Y) ⊕ Z)) ⟩
-    π′ ∘ F.₁ (Push.₁ S.assocˡ) ∘ L′ ∘ φ ∘ (φ ∘ fo ⊗₁ fo) ⊗₁ fo              ≈⟨ refl⟩∘⟨ pullˡ (sym F.homomorphism) ⟩
-    π′ ∘ F.₁ (Push.₁ S.assocˡ ∘′ loop) ∘ φ ∘ (φ ∘ fo ⊗₁ fo) ⊗₁ fo           ≈⟨ pushˡ (sym (π∘l (X ⊕ (Y ⊕ Z)))) ⟩
-    π′ ∘ L′ ∘ F.₁ (Push.₁ S.assocˡ ∘′ loop) ∘ φ ∘ (φ ∘ fo ⊗₁ fo) ⊗₁ fo      ≈⟨ refl⟩∘⟨ pullˡ (sym F.homomorphism) ⟩
-    π′ ∘ F.₁ (loop ∘′ Push.₁ S.assocˡ ∘′ loop) ∘ φ ∘ (φ ∘ fo ⊗₁ fo) ⊗₁ fo   ≈⟨ refl⟩∘⟨ F.F-resp-≈ (loop∘push∘loop S.assocˡ (entire maps-MC.associator.from)) ⟩∘⟨refl ⟩
-    π′ ∘ F.₁ (loop ∘′ Push.₁ S.assocˡ) ∘ φ ∘ (φ ∘ fo ⊗₁ fo) ⊗₁ fo           ≈⟨ refl⟩∘⟨ pushˡ F.homomorphism ⟩
-    π′ ∘ L′ ∘ F.₁ (Push.₁ S.assocˡ) ∘ φ ∘ (φ ∘ fo ⊗₁ fo) ⊗₁ fo              ≈⟨ pullˡ (π∘l (X ⊕ (Y ⊕ Z))) ⟩
-    π′ ∘ F.₁ (Push.₁ S.assocˡ) ∘ φ ∘ (φ ∘ fo ⊗₁ fo) ⊗₁ fo                   ≈⟨ refl⟩∘⟨ F.F-resp-≈ Push-assoc ⟩∘⟨ pushʳ split₁ˡ ⟩
-    π′ ∘ F.₁ BWD.associator.from ∘ (φ ∘ φ ⊗₁ id) ∘ (fo ⊗₁ fo) ⊗₁ fo         ≈⟨ refl⟩∘⟨ extendʳ F.associativity ⟩
-    π′ ∘ φ ∘ (id ⊗₁ φ ∘ 𝒟.associator.from) ∘ (fo ⊗₁ fo) ⊗₁ fo               ≈⟨ refl⟩∘⟨ refl⟩∘⟨ pullʳ 𝒟.assoc-commute-from ⟩
-    π′ ∘ φ ∘ id ⊗₁ φ ∘ fo ⊗₁ (fo ⊗₁ fo) ∘ 𝒟.associator.from                 ≈⟨ refl⟩∘⟨ refl⟩∘⟨ pullˡ merge₂ˡ ⟩
-    π′ ∘ φ ∘ fo ⊗₁ (φ ∘ fo ⊗₁ fo) ∘ 𝒟.associator.from                       ≈⟨ pushˡ (sym (π∘l (X ⊕ (Y ⊕ Z)))) ⟩
-    π′ ∘ L′ ∘ φ ∘ fo ⊗₁ (φ ∘ fo ⊗₁ fo) ∘ 𝒟.associator.from                  ≈⟨ refl⟩∘⟨ F.F-resp-≈ loop⊞loop ⟩∘⟨refl ⟨
-    π′ ∘ F.₁ (loop ⊞₁ loop) ∘ φ ∘ fo ⊗₁ (φ ∘ fo ⊗₁ fo) ∘ 𝒟.associator.from  ≈⟨ refl⟩∘⟨ extendʳ (F.⊗-homo.sym-commute _) ⟩
-    π′ ∘ φ ∘ L′ ⊗₁ L′ ∘ fo ⊗₁ (φ ∘ fo ⊗₁ fo) ∘ 𝒟.associator.from            ≈⟨ refl⟩∘⟨ refl⟩∘⟨ pullˡ (sym ⊗-distrib-over-∘) ⟩
-    π′ ∘ φ ∘ (L′ ∘ fo) ⊗₁ (L′ ∘ φ ∘ fo ⊗₁ fo) ∘ 𝒟.associator.from           ≈⟨ refl⟩∘⟨ refl⟩∘⟨ l∘forget X ⟩⊗⟨ pushˡ (sym (forget∘π (Y ⊕ Z))) ⟩∘⟨refl ⟩
-    π′ ∘ φ ∘ fo ⊗₁ (fo ∘ π′ ∘ φ ∘ fo ⊗₁ fo) ∘ 𝒟.associator.from             ≈⟨ pushʳ (pushʳ (pushˡ split₂ʳ)) ⟩
-    (π′ ∘ φ ∘ fo ⊗₁ fo) ∘ id ⊗₁ (π′ ∘ φ ∘ fo ⊗₁ fo) ∘ 𝒟.associator.from     ∎
-  where
-    open Shorthands
-    open ⊗-Reasoning 𝒟.monoidal
-    open ⇒-Reasoning 𝒟.U
-    open 𝒟.Equiv
-
-unitaryˡ
-    : {X : 𝒞.Obj}
-    → Merge.₁ maps-MC.unitorˡ.from ∘ η maps-MC.unit X ∘ ε ⊗₁ id ≈ 𝒟.unitorˡ.from
-unitaryˡ {X} = begin
-    (π′ ∘ F.₁ (Push.₁ S.π₂) ∘ fo) ∘ (π′ ∘ φ ∘ fo ⊗₁ fo) ∘ ε ⊗₁ id     ≈⟨ pullʳ (pullʳ (extendʳ (pullˡ (forget∘π (S.𝟘 ⊕ X))))) ⟩
-    π′ ∘ F.₁ (Push.₁ S.π₂) ∘ L′ ∘ (φ ∘ fo ⊗₁ fo) ∘ ε ⊗₁ id            ≈⟨ refl⟩∘⟨ refl⟩∘⟨ refl⟩∘⟨ pullʳ merge₁ʳ ⟩
-    π′ ∘ F.₁ (Push.₁ S.π₂) ∘ L′ ∘ φ ∘ (fo ∘ ε) ⊗₁ fo                  ≈⟨ refl⟩∘⟨ refl⟩∘⟨ refl⟩∘⟨ refl⟩∘⟨ pullˡ (forget∘π S.𝟘) ⟩⊗⟨ sym (l∘forget X) ⟩
-    π′ ∘ F.₁ (Push.₁ S.π₂) ∘ L′ ∘ φ ∘ (L′ ∘ F.ε) ⊗₁ (L′ ∘ fo)         ≈⟨ refl⟩∘⟨ refl⟩∘⟨ refl⟩∘⟨ refl⟩∘⟨ ⊗-distrib-over-∘ ⟩
-    π′ ∘ F.₁ (Push.₁ S.π₂) ∘ L′ ∘ φ ∘ L′ ⊗₁ L′ ∘ F.ε ⊗₁ fo            ≈⟨ refl⟩∘⟨ refl⟩∘⟨ refl⟩∘⟨ extendʳ (F.⊗-homo.commute _) ⟩
-    π′ ∘ F.₁ (Push.₁ S.π₂) ∘ L′ ∘ F.₁ (loop ⊞₁ loop) ∘ φ ∘ F.ε ⊗₁ fo  ≈⟨ refl⟩∘⟨ refl⟩∘⟨ refl⟩∘⟨ F.F-resp-≈ loop⊞loop ⟩∘⟨refl ⟩
-    π′ ∘ F.₁ (Push.₁ S.π₂) ∘ L′ ∘ L′ ∘ φ ∘ F.ε ⊗₁ fo                  ≈⟨ refl⟩∘⟨ refl⟩∘⟨ pullˡ (l∘l (S.𝟘 ⊕ X)) ⟩
-    π′ ∘ F.₁ (Push.₁ S.π₂) ∘ L′ ∘ φ ∘ F.ε ⊗₁ fo                       ≈⟨ refl⟩∘⟨ pullˡ (sym F.homomorphism) ⟩
-    π′ ∘ F.₁ (Push.₁ S.π₂ ∘′ loop) ∘ φ ∘ F.ε ⊗₁ fo                    ≈⟨ pushˡ (sym (π∘l X)) ⟩
-    π′ ∘ L′ ∘ F.₁ (Push.₁ S.π₂ ∘′ loop) ∘ φ ∘ F.ε ⊗₁ fo               ≈⟨ refl⟩∘⟨ extendʳ ([ F.F ]-resp-square (loop∘push∘loop S.π₂ (entire maps-MC.unitorˡ.from))) ⟩
-    π′ ∘ L′ ∘ F.₁ (Push.₁ S.π₂) ∘ φ ∘ F.ε ⊗₁ fo                       ≈⟨ pullˡ (π∘l X) ⟩
-    π′ ∘ F.₁ (Push.₁ S.π₂) ∘ φ ∘ F.ε ⊗₁ fo                            ≈⟨ refl⟩∘⟨ F.F-resp-≈ Push-π₂ ⟩∘⟨ pushʳ serialize₁₂ ⟩
-    π′ ∘ F.₁ BWD.unitorˡ.from ∘ (φ ∘ F.ε ⊗₁ id) ∘ id ⊗₁ fo            ≈⟨ refl⟩∘⟨ pullˡ F.unitaryˡ ⟩
-    π′ ∘ 𝒟.unitorˡ.from ∘ id ⊗₁ fo                                    ≈⟨ refl⟩∘⟨ 𝒟.unitorˡ-commute-from ⟩
-    π′ ∘ fo ∘ 𝒟.unitorˡ.from                                          ≈⟨ cancelˡ (π∘forget X) ⟩
-    𝒟.unitorˡ.from                                                    ∎
-  where
-    open Shorthands
-    open ⊗-Reasoning 𝒟.monoidal
-    open ⇒-Reasoning 𝒟.U
-    open 𝒟.Equiv
-
-unitaryʳ
-    : {X : 𝒞.Obj}
-    → Merge.₁ maps-MC.unitorʳ.from ∘ η X maps-MC.unit ∘ id ⊗₁ ε ≈ 𝒟.unitorʳ.from
-unitaryʳ {X} = begin
-    (π′ ∘ F.₁ (Push.₁ S.π₁) ∘ fo) ∘ (π′ ∘ φ ∘ fo ⊗₁ fo) ∘ id ⊗₁ ε     ≈⟨ pullʳ (pullʳ (extendʳ (pullˡ (forget∘π (X ⊕ S.𝟘))))) ⟩
-    π′ ∘ F.₁ (Push.₁ S.π₁) ∘ L′ ∘ (φ ∘ fo ⊗₁ fo) ∘ id ⊗₁ ε            ≈⟨ refl⟩∘⟨ refl⟩∘⟨ refl⟩∘⟨ pullʳ merge₂ʳ ⟩
-    π′ ∘ F.₁ (Push.₁ S.π₁) ∘ L′ ∘ φ ∘ fo ⊗₁ (fo ∘ ε)                  ≈⟨ refl⟩∘⟨ refl⟩∘⟨ refl⟩∘⟨ refl⟩∘⟨ sym (l∘forget X) ⟩⊗⟨ pullˡ (forget∘π S.𝟘) ⟩
-    π′ ∘ F.₁ (Push.₁ S.π₁) ∘ L′ ∘ φ ∘ (L′ ∘ fo) ⊗₁ (L′ ∘ F.ε)         ≈⟨ refl⟩∘⟨ refl⟩∘⟨ refl⟩∘⟨ refl⟩∘⟨ ⊗-distrib-over-∘ ⟩
-    π′ ∘ F.₁ (Push.₁ S.π₁) ∘ L′ ∘ φ ∘ L′ ⊗₁ L′ ∘ fo ⊗₁ F.ε            ≈⟨ refl⟩∘⟨ refl⟩∘⟨ refl⟩∘⟨ extendʳ (F.⊗-homo.commute _) ⟩
-    π′ ∘ F.₁ (Push.₁ S.π₁) ∘ L′ ∘ F.₁ (loop ⊞₁ loop) ∘ φ ∘ fo ⊗₁ F.ε  ≈⟨ refl⟩∘⟨ refl⟩∘⟨ refl⟩∘⟨ F.F-resp-≈ loop⊞loop ⟩∘⟨refl ⟩
-    π′ ∘ F.₁ (Push.₁ S.π₁) ∘ L′ ∘ L′ ∘ φ ∘ fo ⊗₁ F.ε                  ≈⟨ refl⟩∘⟨ refl⟩∘⟨ pullˡ (l∘l (X ⊕ S.𝟘)) ⟩
-    π′ ∘ F.₁ (Push.₁ S.π₁) ∘ L′ ∘ φ ∘ fo ⊗₁ F.ε                       ≈⟨ refl⟩∘⟨ pullˡ (sym F.homomorphism) ⟩
-    π′ ∘ F.₁ (Push.₁ S.π₁ ∘′ loop) ∘ φ ∘ fo ⊗₁ F.ε                    ≈⟨ pushˡ (sym (π∘l X)) ⟩
-    π′ ∘ L′ ∘ F.₁ (Push.₁ S.π₁ ∘′ loop) ∘ φ ∘ fo ⊗₁ F.ε               ≈⟨ refl⟩∘⟨ extendʳ ([ F.F ]-resp-square (loop∘push∘loop S.π₁ (entire maps-MC.unitorʳ.from))) ⟩
-    π′ ∘ L′ ∘ F.₁ (Push.₁ S.π₁) ∘ φ ∘ fo ⊗₁ F.ε                       ≈⟨ pullˡ (π∘l X) ⟩
-    π′ ∘ F.₁ (Push.₁ S.π₁) ∘ φ ∘ fo ⊗₁ F.ε                            ≈⟨ refl⟩∘⟨ F.F-resp-≈ Push-π₁ ⟩∘⟨ pushʳ serialize₂₁ ⟩
-    π′ ∘ F.₁ BWD.unitorʳ.from ∘ (φ ∘ id ⊗₁ F.ε) ∘ fo ⊗₁ id            ≈⟨ refl⟩∘⟨ pullˡ F.unitaryʳ ⟩
-    π′ ∘ 𝒟.unitorʳ.from ∘ fo ⊗₁ id                                    ≈⟨ refl⟩∘⟨ 𝒟.unitorʳ-commute-from ⟩
-    π′ ∘ fo ∘ 𝒟.unitorʳ.from                                          ≈⟨ cancelˡ (π∘forget X) ⟩
-    𝒟.unitorʳ.from                                                    ∎
-  where
-    open Shorthands
-    open ⊗-Reasoning 𝒟.monoidal
-    open ⇒-Reasoning 𝒟.U
-    open 𝒟.Equiv
-
-Merge-IsMF : IsMonoidalFunctor S.maps-MC 𝒟 Merge
-Merge-IsMF = record
-    { ε = ε
-    ; ⊗-homo = ⊗-homo
-    ; associativity = associativity
-    ; unitaryˡ = unitaryˡ
-    ; unitaryʳ = unitaryʳ
-    }
-
-Merge-MF : MonoidalFunctor S.maps-MC 𝒟
-Merge-MF = record
-    { F = Merge
-    ; isMonoidal = Merge-IsMF
-    }