Construct monoidal merge functor

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+{-# OPTIONS --without-K --safe #-}
+
+open import Categories.Category using (Category)
+open import Categories.Functor using (Functor; _∘F_)
+open import Category.Dagger.Semiadditive using (SemiadditiveDagger; IdempotentSemiadditiveDagger)
+open import Category.KaroubiComplete using (KaroubiComplete)
+open import Data.WiringDiagram.Balanced using (BWD)
+open import Level using (Level)
+
+module Data.WiringDiagram.Looped.Core
+    {o ℓ e o′ ℓ′ e′ : Level}
+    {𝒞 : Category o ℓ e}
+    {𝒟 : Category o′ ℓ′ e′}
+    {S : IdempotentSemiadditiveDagger 𝒞}
+    (let module S = IdempotentSemiadditiveDagger S)
+    (let S′ = S.semiadditiveDagger)
+    (karoubiComplete : KaroubiComplete 𝒟)
+    (F : Functor (BWD S′) 𝒟)
+  where
+
+import Categories.Morphism.Idempotent as Idempotent
+import Categories.Morphism.Reasoning as ⇒-Reasoning
+
+open import Categories.Category using (Category)
+open import Categories.Functor.Properties using ([_]-resp-∘)
+open import Category.Dagger.2-Poset using (Dagger-2-Poset; Maps; Map)
+open import Data.WiringDiagram.Balanced S′ using (Include; Push; Pull)
+open import Data.WiringDiagram.Core S′ using (loop; id-⧈; _□_)
+open import Data.WiringDiagram.Equalities S using (loop∘loop; loop∘push∘loop; loop∘pull∘loop)
+
+module BWD = Category (BWD S′)
+module F = Functor F
+module 𝒞 = Category 𝒞
+module 𝒟 = Category 𝒟
+
+open Category using (op)
+open Idempotent 𝒟 using (IsSplitIdempotent)
+
+module _ (A : 𝒞.Obj) where
+
+  open KaroubiComplete karoubiComplete using (split)
+  open IsSplitIdempotent (split ([ F ]-resp-∘ (loop∘loop {A})))
+
+  Unlooped Looped : 𝒟.Obj
+  Unlooped = F.₀ A
+  Looped = obj
+
+  L : Unlooped 𝒟.⇒ Unlooped
+  L = F.₁ loop
+
+  π : Unlooped 𝒟.⇒ Looped
+  π = retract
+
+  forget : Looped 𝒟.⇒ Unlooped
+  forget = section
+
+  forget∘π : forget 𝒟.∘ π 𝒟.≈ L
+  forget∘π = splits
+
+  π∘forget : π 𝒟.∘ forget 𝒟.≈ 𝒟.id
+  π∘forget = retracts
+
+  π∘l : π 𝒟.∘ L 𝒟.≈ π
+  π∘l = retract-absorb
+
+  l∘forget : L 𝒟.∘ forget 𝒟.≈ forget
+  l∘forget = section-absorb
+
+  l∘l : L 𝒟.∘ L 𝒟.≈ L
+  l∘l = [ F ]-resp-∘ loop∘loop
+
+module Push = Functor Push
+module Pull = Functor Pull
+
+S-≤ : Dagger-2-Poset
+S-≤ = S.dagger-2-poset
+
+Merge : Functor (Maps S-≤) 𝒟
+Merge = record
+    { F₀ = Looped
+    ; F₁ = λ {A} {B} f → π B ∘ F.₁ (Push.₁ (map f)) ∘ forget A
+    ; identity = iden
+    ; homomorphism = λ {f = f} {g} → homo {f = f} {g}
+    ; F-resp-≈ = resp
+    }
+  where
+    open Map
+    open Category 𝒟 using (_∘_)
+    open 𝒟.HomReasoning
+    open ⇒-Reasoning 𝒟
+    iden : {A : 𝒞.Obj} → π A ∘ F.₁ (Push.₁ 𝒞.id) ∘ forget A 𝒟.≈ 𝒟.id
+    iden {A} = begin
+        π A ∘ F.₁ (Push.₁ 𝒞.id) ∘ forget A  ≈⟨ refl⟩∘⟨ F.F-resp-≈ Push.identity ⟩∘⟨refl ⟩
+        π A ∘ F.₁ BWD.id ∘ forget A         ≈⟨ refl⟩∘⟨ elimˡ F.identity ⟩
+        π A ∘ forget A                      ≈⟨ π∘forget A ⟩
+        𝒟.id                                ∎
+    homo
+        : {X Y Z : 𝒞.Obj}
+          {f : Map S-≤ X Y}
+          {g : Map S-≤ Y Z}
+        → π Z ∘ F.₁ (Push.₁ (map g 𝒞.∘ map f)) ∘ forget X 𝒟.≈ (π Z ∘ F.₁ (Push.₁ (map g)) ∘ forget Y) ∘ π Y ∘ F.₁ (Push.₁ (map f)) ∘ forget X
+    homo {X} {Y} {Z} {f′} {g′} = begin
+        π Z ∘ F.₁ (Push.₁ (g 𝒞.∘ f)) ∘ forget X                                 ≈⟨ refl⟩∘⟨ F.F-resp-≈ Push.homomorphism ⟩∘⟨refl ⟩
+        π Z ∘ F.₁ (Push.₁ g BWD.∘ Push.₁ f) ∘ forget X                          ≈⟨ refl⟩∘⟨ pushˡ F.homomorphism ⟩
+        π Z ∘ F.₁ (Push.₁ g) ∘ F.₁ (Push.₁ f) ∘ forget X                        ≈⟨ pushˡ (𝒟.Equiv.sym (π∘l Z)) ⟩
+        π Z ∘ L Z ∘ F.₁ (Push.₁ g) ∘ F.₁ (Push.₁ f) ∘ forget X                  ≈⟨ refl⟩∘⟨ pullˡ (𝒟.Equiv.sym F.homomorphism) ⟩
+        π Z ∘ F.₁ (loop BWD.∘ Push.₁ g) ∘ F.₁ (Push.₁ f) ∘ forget X             ≈⟨ refl⟩∘⟨ F.F-resp-≈ (loop∘push∘loop g (entire g′)) ⟩∘⟨refl ⟨
+        π Z ∘ F.₁ (loop BWD.∘ Push.₁ g BWD.∘ loop) ∘ F.₁ (Push.₁ f) ∘ forget X  ≈⟨ refl⟩∘⟨ pushˡ F.homomorphism ⟩
+        π Z ∘ L Z ∘ F.₁ (Push.₁ g BWD.∘ loop) ∘ F.₁ (Push.₁ f) ∘ forget X       ≈⟨ refl⟩∘⟨ refl⟩∘⟨ pushˡ F.homomorphism ⟩
+        π Z ∘ L Z ∘ F.₁ (Push.₁ g) ∘ L Y ∘ F.₁ (Push.₁ f) ∘ forget X            ≈⟨ pullˡ (π∘l Z) ⟩
+        π Z ∘ F.₁ (Push.₁ g) ∘ L Y ∘ F.₁ (Push.₁ f) ∘ forget X                  ≈⟨ pushʳ (pushʳ (pushˡ (𝒟.Equiv.sym (forget∘π Y)))) ⟩
+        (π Z ∘ F.₁ (Push.₁ g) ∘ forget Y) ∘ π Y ∘ F.₁ (Push.₁ f) ∘ forget X     ∎
+      where
+        f : X 𝒞.⇒ Y
+        f = map f′
+        g : Y 𝒞.⇒ Z
+        g = map g′
+    resp : {A B : 𝒞.Obj} {f g : A 𝒞.⇒ B} → f 𝒞.≈ g → π B ∘ F.₁ (Push.₁ f) ∘ forget A 𝒟.≈ π B ∘ F.₁ (Push.₁ g) ∘ forget A
+    resp {A} {B} {f} {g} f≈g = refl⟩∘⟨ F.F-resp-≈ (Push.F-resp-≈ f≈g) ⟩∘⟨refl
+
+Split : Functor (op (Maps S-≤)) 𝒟
+Split = record
+    { F₀ = Looped
+    ; F₁ = λ {A} {B} f → π B ∘ F.₁ (Pull.₁ (map f)) ∘ forget A
+    ; identity = iden
+    ; homomorphism = λ {f = f} {g} → homo {f = f} {g}
+    ; F-resp-≈ = resp
+    }
+  where
+    open Map
+    open Category 𝒟 using (_∘_)
+    open 𝒟.HomReasoning
+    open ⇒-Reasoning 𝒟
+    iden : {A : 𝒞.Obj} → π A ∘ F.₁ (Pull.₁ 𝒞.id) ∘ forget A 𝒟.≈ 𝒟.id
+    iden {A} = begin
+        π A ∘ F.₁ (Pull.₁ 𝒞.id) ∘ forget A  ≈⟨ refl⟩∘⟨ F.F-resp-≈ Pull.identity ⟩∘⟨refl ⟩
+        π A ∘ F.₁ BWD.id ∘ forget A         ≈⟨ refl⟩∘⟨ elimˡ F.identity ⟩
+        π A ∘ forget A                      ≈⟨ π∘forget A ⟩
+        𝒟.id                                ∎
+    homo
+        : {X Y Z : 𝒞.Obj}
+          {f : Map S-≤ Y X}
+          {g : Map S-≤ Z Y}
+        → π Z ∘ F.₁ (Pull.₁ (map f 𝒞.∘ map g)) ∘ forget X 𝒟.≈ (π Z ∘ F.₁ (Pull.₁ (map g)) ∘ forget Y) ∘ π Y ∘ F.₁ (Pull.₁ (map f)) ∘ forget X
+    homo {X} {Y} {Z} {f′} {g′} = begin
+        π Z ∘ F.₁ (Pull.₁ (f 𝒞.∘ g)) ∘ forget X                                 ≈⟨ refl⟩∘⟨ F.F-resp-≈ Pull.homomorphism ⟩∘⟨refl ⟩
+        π Z ∘ F.₁ (Pull.₁ g BWD.∘ Pull.₁ f) ∘ forget X                          ≈⟨ refl⟩∘⟨ pushˡ F.homomorphism ⟩
+        π Z ∘ F.₁ (Pull.₁ g) ∘ F.₁ (Pull.₁ f) ∘ forget X                        ≈⟨ refl⟩∘⟨ refl⟩∘⟨ refl⟩∘⟨ l∘forget X ⟨
+        π Z ∘ F.₁ (Pull.₁ g) ∘ F.₁ (Pull.₁ f) ∘ L X ∘ forget X                  ≈⟨ refl⟩∘⟨ refl⟩∘⟨ pullˡ (𝒟.Equiv.sym F.homomorphism) ⟩
+        π Z ∘ F.₁ (Pull.₁ g) ∘ F.₁ (Pull.₁ f BWD.∘ loop) ∘ forget X             ≈⟨ refl⟩∘⟨ refl⟩∘⟨ F.F-resp-≈ (loop∘pull∘loop f (functional f′)) ⟩∘⟨refl ⟨
+        π Z ∘ F.₁ (Pull.₁ g) ∘ F.₁ (loop BWD.∘ Pull.₁ f BWD.∘ loop) ∘ forget X  ≈⟨ refl⟩∘⟨ refl⟩∘⟨ pushˡ F.homomorphism ⟩
+        π Z ∘ F.₁ (Pull.₁ g) ∘ L Y ∘ F.₁ (Pull.₁ f BWD.∘ loop) ∘ forget X       ≈⟨ refl⟩∘⟨ refl⟩∘⟨ refl⟩∘⟨ pushˡ F.homomorphism ⟩
+        π Z ∘ F.₁ (Pull.₁ g) ∘ L Y ∘ F.₁ (Pull.₁ f) ∘ L X ∘ forget X            ≈⟨ refl⟩∘⟨ refl⟩∘⟨ refl⟩∘⟨ refl⟩∘⟨ l∘forget X ⟩
+        π Z ∘ F.₁ (Pull.₁ g) ∘ L Y ∘ F.₁ (Pull.₁ f) ∘ forget X                  ≈⟨ pushʳ (pushʳ (pushˡ (𝒟.Equiv.sym (forget∘π Y)))) ⟩
+        (π Z ∘ F.₁ (Pull.₁ g) ∘ forget Y) ∘ π Y ∘ F.₁ (Pull.₁ f) ∘ forget X     ∎
+      where
+        f : Y 𝒞.⇒ X
+        f = map f′
+        g : Z 𝒞.⇒ Y
+        g = map g′
+    resp : {A B : 𝒞.Obj} {f g : B 𝒞.⇒ A} → f 𝒞.≈ g → π B ∘ F.₁ (Pull.₁ f) ∘ forget A 𝒟.≈ π B ∘ F.₁ (Pull.₁ g) ∘ forget A
+    resp {A} {B} {f} {g} f≈g = refl⟩∘⟨ F.F-resp-≈ (Pull.F-resp-≈ f≈g) ⟩∘⟨refl