<!--
```agda
{-# OPTIONS --allow-unsolved-metas #-}
open import Level
open import Categories.Category
open import Categories.Monad
open import Categories.Category.Distributive
open import Categories.Category.Extensive.Bundle
open import Categories.Category.Extensive
open import Categories.Category.BinaryProducts
open import Categories.Category.Cocartesian
open import Categories.Category.Cartesian
open import Categories.Category.Cartesian.Bundle
open import Categories.Object.Terminal
open import Categories.Object.Initial
open import Categories.Object.Coproduct
open import Categories.Category.Construction.F-Coalgebras
open import Categories.Category.Construction.F-Algebras
open import Categories.Functor.Coalgebra
open import Categories.Functor
open import Categories.Functor.Algebra
open import Categories.Monad.Construction.Kleisli
open import Categories.Category.Construction.F-Coalgebras
open import Categories.NaturalTransformation
open import FinalCoalgebras
import Categories.Morphism as M
import Categories.Morphism.Reasoning as MR
```
-->

## Summary
This file introduces the delay monad ***D***

- [ ] *Proposition 1* Characterization of the delay monad ***D***
- [ ] *Proposition 2* ***D*** is commutative

## Code

```agda
module Monad.Instance.Delay {o ℓ e} (ED : ExtensiveDistributiveCategory o ℓ e) where
  open ExtensiveDistributiveCategory ED renaming (U to C; id to idC)
  open Cocartesian (Extensive.cocartesian extensive)
  open Cartesian (ExtensiveDistributiveCategory.cartesian ED)
  open BinaryProducts products

  CC : CartesianCategory o ℓ e
  CC = record { U = C ; cartesian = (ExtensiveDistributiveCategory.cartesian ED) }

  open import Categories.Object.NaturalNumbers.Parametrized CC
  open import Categories.Object.NaturalNumbers.Properties.F-Algebras using (PNNO⇒Initial₂; PNNO-Algebra)

  open M C
  open MR C
  open Equiv
  open HomReasoning
  open CoLambek
  open F-Coalgebra-Morphism
```
### *Proposition 1*: Characterization of the delay monad ***D***

```agda
  delayF : Obj → Endofunctor C
  delayF Y = record
    { F₀ = Y +_
    ; F₁ = idC +₁_
    ; identity = CC.coproduct.unique id-comm-sym id-comm-sym
    ; homomorphism = ⟺ (+₁∘+₁ ○ +₁-cong₂ identity² refl) 
    ; F-resp-≈ = +₁-cong₂ refl
    }

  record DelayM : Set (o ⊔ ℓ ⊔ e) where
    field
      algebras : ∀ (A : Obj) → Terminal (F-Coalgebras (delayF A))
    
    module D A = Functor (delayF A)

    module _ (X : Obj) where
      open Terminal (algebras X) using (⊤; !; !-unique)
      open F-Coalgebra ⊤ renaming (A to DX)

      D₀ = DX

      out-≅ : DX ≅ X + DX
      out-≅ = colambek {F = delayF X} (algebras X)

      -- note: out-≅.from ≡ ⊤.α
      open _≅_ out-≅ using () renaming (to to out⁻¹; from to out) public

      now : X ⇒ DX
      now = out⁻¹ ∘ i₁

      later : DX ⇒ DX
      later = out⁻¹ ∘ i₂

      -- TODO inline
      unitlaw : out ∘ now ≈ i₁
      unitlaw = cancelˡ (_≅_.isoʳ out-≅)

      module _ {Y : Obj} where 
        coit : Y ⇒ X + Y → Y ⇒ DX
        coit f = F-Coalgebra-Morphism.f (! {A = record { A = Y ; α = f }})

        coit-commutes : ∀ (f : Y ⇒ X + Y) → out ∘ (coit f) ≈ (idC +₁ coit f) ∘ f
        coit-commutes f = F-Coalgebra-Morphism.commutes (! {A = record { A = Y ; α = f }})

      module _ (ℕ : ParametrizedNNO) where
        open ParametrizedNNO ℕ
      
        iso : X × N ≅ X + X × N
        iso = Lambek.lambek (record { ⊥ = PNNO-Algebra CC coproducts X N z s ; ⊥-is-initial = PNNO⇒Initial₂ CC coproducts ℕ X })

        ι : X × N ⇒ DX
        ι = f (! {A = record { A = X × N ; α = _≅_.from iso }})

        
    monad : Monad C
    monad = Kleisli⇒Monad C (record
      { F₀ = D₀ 
      ; unit = λ {X} → now X
      ; extend = extend
      ; identityʳ = λ {X} {Y} {f} → begin 
        extend f ∘ now X                                          ≈⟨ (insertˡ (_≅_.isoˡ (out-≅ Y))) ⟩∘⟨refl ⟩ 
        (out⁻¹ Y ∘ out Y ∘ extend f) ∘ now X                      ≈⟨ (refl⟩∘⟨ (extendlaw f)) ⟩∘⟨refl ⟩
        (out⁻¹ Y ∘ [ out Y ∘ f , i₂ ∘ extend f ] ∘ out X) ∘ now X ≈⟨ pullʳ (pullʳ (unitlaw X)) ⟩
        out⁻¹ Y ∘ [ out Y ∘ f , i₂ ∘ extend f ] ∘ i₁              ≈⟨ refl⟩∘⟨ inject₁ ⟩
        out⁻¹ Y ∘ out Y ∘ f                                       ≈⟨ cancelˡ (_≅_.isoˡ (out-≅ Y)) ⟩
        f                                                         ∎
      ; identityˡ = λ {X} → Terminal.⊤-id (algebras X) (record { f = extend (now X) ; commutes = begin 
        out X ∘ extend (now X)                                                                                             ≈⟨ pullˡ ((F-Coalgebra-Morphism.commutes (Terminal.! (algebras X) {A = alg (now X)}))) ⟩ 
        ((idC +₁ (F-Coalgebra-Morphism.f (Terminal.! (algebras X) {A = alg (now X)}))) ∘ F-Coalgebra.α (alg (now X))) ∘ i₁ ≈⟨ pullʳ inject₁ ⟩
        (idC +₁ (F-Coalgebra-Morphism.f (Terminal.! (algebras X) {A = alg (now X)}))) 
        ∘ [ [ i₁ , i₂ ∘ i₂ ] ∘ (out X ∘ (now X)) , i₂ ∘ i₁ ] ∘ out X                                                       ≈⟨ refl⟩∘⟨ []-cong₂ ((refl⟩∘⟨ (unitlaw X)) ○ inject₁) refl ⟩∘⟨refl ⟩
        (idC +₁ (F-Coalgebra-Morphism.f (Terminal.! (algebras X) {A = alg (now X)}))) ∘ [ i₁ , i₂ ∘ i₁ ] ∘ out X           ≈⟨ pullˡ ∘[] ⟩
        [ (idC +₁ (F-Coalgebra-Morphism.f (Terminal.! (algebras X) {A = alg (now X)}))) ∘ i₁ 
        , (idC +₁ (F-Coalgebra-Morphism.f (Terminal.! (algebras X) {A = alg (now X)}))) ∘ i₂ ∘ i₁ ] ∘ out X                ≈⟨ ([]-cong₂ +₁∘i₁ (pullˡ +₁∘i₂)) ⟩∘⟨refl ⟩
        [ i₁ ∘ idC , (i₂ ∘ (F-Coalgebra-Morphism.f (Terminal.! (algebras X) {A = alg (now X)}))) ∘ i₁ ] ∘ out X            ≈⟨ ([]-cong₂ refl assoc) ⟩∘⟨refl ⟩
        [ i₁ ∘ idC , i₂ ∘ (extend (now X)) ] ∘ out X                                                                       ≈˘⟨ []∘+₁ ⟩∘⟨refl ⟩
        ([ i₁ , i₂ ] ∘ (idC +₁ extend (now X))) ∘ out X                                                                    ≈⟨ (elimˡ +-η) ⟩∘⟨refl ⟩
        (idC +₁ extend (now X)) ∘ out X                                                                                    ∎ })
      ; assoc = λ {X} {Y} {Z} {g} {h} → {!   !}
      
      -- begin 
      --   extend (extend h ∘ g) ≈⟨ insertˡ (_≅_.isoˡ (out-≅ Z)) ⟩ 
      --   out⁻¹ Z ∘ out Z ∘ extend (extend h ∘ g) ≈⟨ refl⟩∘⟨ (pullˡ (commutes (! (algebras Z)))) ⟩ 
      --   out⁻¹ Z ∘ ((idC +₁ (f (! (algebras Z)))) ∘ F-Coalgebra.α (alg (extend h ∘ g))) ∘ i₁ ≈⟨ refl⟩∘⟨ (pullʳ inject₁) ⟩ 
      --   out⁻¹ Z ∘ (idC +₁ (f (! (algebras Z)))) ∘ [ [ i₁ , i₂ ∘ i₂ ] ∘ out Z ∘ extend h ∘ g , i₂ ∘ i₁ ] ∘ out X ≈⟨ refl⟩∘⟨ pullˡ ∘[] ⟩
      --   out⁻¹ Z ∘ [ (idC +₁ (f (! (algebras Z)))) ∘ [ i₁ , i₂ ∘ i₂ ] ∘ out Z ∘ extend h ∘ g , (idC +₁ (f (! (algebras Z)))) ∘ i₂ ∘ i₁ ] ∘ out X ≈⟨ refl⟩∘⟨ (([]-cong₂ (pullˡ ∘[]) (pullˡ +₁∘i₂)) ⟩∘⟨refl) ⟩
      --   out⁻¹ Z ∘ [ [ (idC +₁ (f (! (algebras Z)))) ∘ i₁ , (idC +₁ (f (! (algebras Z)))) ∘ i₂ ∘ i₂ ] ∘ out Z ∘ extend h ∘ g , (i₂ ∘ (f (! (algebras Z)))) ∘ i₁ ] ∘ out X ≈⟨ refl⟩∘⟨ (([]-cong₂ (([]-cong₂ (+₁∘i₁ ○ identityʳ) (pullˡ +₁∘i₂)) ⟩∘⟨refl) refl) ⟩∘⟨refl) ⟩
      --   out⁻¹ Z ∘ [ [ i₁ , (i₂ ∘ (f (! (algebras Z)))) ∘ i₂ ] ∘ out Z ∘ extend h ∘ g 
      --             , (i₂ ∘ (f (! (algebras Z)))) ∘ i₁ 
      --             ] ∘ out X ≈⟨ refl⟩∘⟨ (([]-cong₂ (refl⟩∘⟨ (pullˡ (pullˡ (commutes (! (algebras Z)))))) refl) ⟩∘⟨refl) ⟩
      --   out⁻¹ Z ∘ [ [ i₁ , (i₂ ∘ (f (! (algebras Z)))) ∘ i₂ ] ∘ (((idC +₁ f (! (algebras Z))) ∘ F-Coalgebra.α (alg h)) ∘ i₁) ∘ g , (i₂ ∘ (f (! (algebras Z)))) ∘ i₁ ] ∘ out X ≈⟨ refl⟩∘⟨ (([]-cong₂ (refl⟩∘⟨ ((pullʳ inject₁) ⟩∘⟨refl)) refl) ⟩∘⟨refl) ⟩
      --   out⁻¹ Z ∘ [ [ i₁ , (i₂ ∘ (f (! (algebras Z)))) ∘ i₂ ] ∘ ((idC +₁ f (! (algebras Z))) ∘ [ [ i₁ , i₂ ∘ i₂ ] ∘ out Z ∘ h , i₂ ∘ i₁ ] ∘ out Y) ∘ g , (i₂ ∘ (f (! (algebras Z)))) ∘ i₁ ] ∘ out X ≈⟨ refl⟩∘⟨ (([]-cong₂ (pullˡ (pullˡ []∘+₁)) refl) ⟩∘⟨refl) ⟩
      --   out⁻¹ Z ∘ [ ([ i₁ ∘ idC , ((i₂ ∘ (f (! (algebras Z)))) ∘ i₂) ∘ f (! (algebras Z)) ] ∘ ([ [ i₁ , i₂ ∘ i₂ ] ∘ out Z ∘ h , i₂ ∘ i₁ ] ∘ out Y)) ∘ g , (i₂ ∘ (f (! (algebras Z)))) ∘ i₁ ] ∘ out X ≈⟨ refl⟩∘⟨ (([]-cong₂ ((pullˡ ∘[]) ⟩∘⟨refl) refl) ⟩∘⟨refl) ⟩
      --   out⁻¹ Z ∘ [ ([ [ i₁ ∘ idC , ((i₂ ∘ (f (! (algebras Z)))) ∘ i₂) ∘ f (! (algebras Z)) ] ∘ [ i₁ , i₂ ∘ i₂ ] ∘ out Z ∘ h , [ i₁ ∘ idC , ((i₂ ∘ (f (! (algebras Z)))) ∘ i₂) ∘ f (! (algebras Z)) ] ∘ i₂ ∘ i₁ ] ∘ out Y) ∘ g , (i₂ ∘ (f (! (algebras Z)))) ∘ i₁ ] ∘ out X ≈⟨ refl⟩∘⟨ (([]-cong₂ ((([]-cong₂ (pullˡ ∘[]) (pullˡ inject₂)) ⟩∘⟨refl) ⟩∘⟨refl) refl) ⟩∘⟨refl) ⟩ 
      --   out⁻¹ Z ∘ [ ([ [ [ i₁ ∘ idC , ((i₂ ∘ (f (! (algebras Z)))) ∘ i₂) ∘ f (! (algebras Z)) ] ∘ i₁ , [ i₁ ∘ idC , ((i₂ ∘ (f (! (algebras Z)))) ∘ i₂) ∘ f (! (algebras Z)) ] ∘ i₂ ∘ i₂ ] ∘ out Z ∘ h , (((i₂ ∘ (f (! (algebras Z)))) ∘ i₂) ∘ f (! (algebras Z))) ∘ i₁ ] ∘ out Y) ∘ g , (i₂ ∘ (f (! (algebras Z)))) ∘ i₁ ] ∘ out X ≈⟨ refl⟩∘⟨ (([]-cong₂ ((([]-cong₂ (([]-cong₂ inject₁ (pullˡ inject₂)) ⟩∘⟨refl) refl) ⟩∘⟨refl) ⟩∘⟨refl) refl) ⟩∘⟨refl) ⟩ 
      --   out⁻¹ Z ∘ [ ([ [ i₁ ∘ idC , (((i₂ ∘ (f (! (algebras Z)))) ∘ i₂) ∘ f (! (algebras Z))) ∘ i₂ ] ∘ out Z ∘ h , (((i₂ ∘ (f (! (algebras Z)))) ∘ i₂) ∘ f (! (algebras Z))) ∘ i₁ ] ∘ out Y) ∘ g 
      --             , (i₂ ∘ (f (! (algebras Z)))) ∘ i₁ 
      --             ] ∘ out X ≈⟨ {!   !} ⟩ 
      --   {!   !} ≈⟨ {!   !} ⟩ 
      --   {!   !} ≈˘⟨ {!  _○_ !} ⟩ 
      --   {!   !} ≈˘⟨ {!   !} ⟩ 
      --   {!   !} ≈˘⟨ {!   !} ⟩ 
      --   out⁻¹ Z ∘ [ [ [ i₁ , (i₂ ∘ f (! (algebras Z))) ∘ i₂ ] ∘ out Z ∘ h , (((i₂ ∘ f (! (algebras Z))) ∘ i₁) ∘ f (! (algebras Y))) ∘ i₂ ] ∘ out Y ∘ g 
      --             , (((i₂ ∘ f (! (algebras Z))) ∘ i₁) ∘ f (! (algebras Y))) ∘ i₁ 
      --             ] ∘ out X ≈˘⟨ refl⟩∘⟨ (([]-cong₂ (([]-cong₂ inject₁ (pullˡ inject₂)) ⟩∘⟨refl) refl) ⟩∘⟨refl) ⟩ 
      --   out⁻¹ Z ∘ [ [ [ [ i₁ , (i₂ ∘ f (! (algebras Z))) ∘ i₂ ] ∘ out Z ∘ h , ((i₂ ∘ f (! (algebras Z))) ∘ i₁) ∘ f (! (algebras Y)) ] ∘ i₁ , [ [ i₁ , (i₂ ∘ f (! (algebras Z))) ∘ i₂ ] ∘ out Z ∘ h , ((i₂ ∘ f (! (algebras Z))) ∘ i₁) ∘ f (! (algebras Y)) ] ∘ i₂ ∘ i₂ ] ∘ out Y ∘ g 
      --             , (((i₂ ∘ f (! (algebras Z))) ∘ i₁) ∘ f (! (algebras Y))) ∘ i₁ ] ∘ out X ≈˘⟨ refl⟩∘⟨ (([]-cong₂ (pullˡ ∘[]) (pullˡ inject₂)) ⟩∘⟨refl) ⟩ 
      --   out⁻¹ Z ∘ [ [ [ i₁ , (i₂ ∘ f (! (algebras Z))) ∘ i₂ ] ∘ out Z ∘ h , ((i₂ ∘ f (! (algebras Z))) ∘ i₁) ∘ f (! (algebras Y)) ] ∘ [ i₁ , i₂ ∘ i₂ ] ∘ out Y ∘ g 
      --             , [ [ i₁ , (i₂ ∘ f (! (algebras Z))) ∘ i₂ ] ∘ out Z ∘ h , ((i₂ ∘ f (! (algebras Z))) ∘ i₁) ∘ f (! (algebras Y)) ] ∘ i₂ ∘ i₁ ] ∘ out X ≈˘⟨ refl⟩∘⟨ (pullˡ ∘[]) ⟩ 
      --   out⁻¹ Z ∘ [ [ i₁ , (i₂ ∘ f (! (algebras Z))) ∘ i₂ ] ∘ out Z ∘ h , ((i₂ ∘ f (! (algebras Z))) ∘ i₁) ∘ f (! (algebras Y)) ] ∘ [ [ i₁ , i₂ ∘ i₂ ] ∘ out Y ∘ g , i₂ ∘ i₁ ] ∘ out X ≈˘⟨ refl⟩∘⟨ (([]-cong₂ (([]-cong₂ (+₁∘i₁ ○ identityʳ) (pullˡ +₁∘i₂)) ⟩∘⟨refl) refl) ⟩∘⟨refl) ⟩ 
      --   out⁻¹ Z ∘ [ [ (idC +₁ f (! (algebras Z))) ∘ i₁ , (idC +₁ f (! (algebras Z))) ∘ i₂ ∘ i₂ ] ∘ out Z ∘ h , ((i₂ ∘ f (! (algebras Z))) ∘ i₁) ∘ f (! (algebras Y)) ] ∘ [ [ i₁ , i₂ ∘ i₂ ] ∘ out Y ∘ g , i₂ ∘ i₁ ] ∘ out X ≈˘⟨ refl⟩∘⟨ (([]-cong₂ ((refl⟩∘⟨ identityʳ) ○ (pullˡ ∘[])) (pullˡ (pullˡ +₁∘i₂))) ⟩∘⟨refl) ⟩ 
      --   out⁻¹ Z ∘ [ (idC +₁ f (! (algebras Z))) ∘ ([ i₁ , i₂ ∘ i₂ ] ∘ out Z ∘ h) ∘ idC , (idC +₁ f (! (algebras Z))) ∘ (i₂ ∘ i₁) ∘ f (! (algebras Y)) ] ∘ [ [ i₁ , i₂ ∘ i₂ ] ∘ out Y ∘ g , i₂ ∘ i₁ ] ∘ out X ≈˘⟨ refl⟩∘⟨ (pullˡ ∘[]) ⟩ 
      --   out⁻¹ Z ∘ (idC +₁ f (! (algebras Z))) ∘ [ ([ i₁ , i₂ ∘ i₂ ] ∘ out Z ∘ h) ∘ idC , (i₂ ∘ i₁) ∘ f (! (algebras Y)) ] ∘ [ [ i₁ , i₂ ∘ i₂ ] ∘ out Y ∘ g , i₂ ∘ i₁ ] ∘ out X ≈˘⟨ refl⟩∘⟨ (refl⟩∘⟨ (pullˡ []∘+₁)) ⟩ 
      --   out⁻¹ Z ∘ (idC +₁ f (! (algebras Z))) ∘ [ [ i₁ , i₂ ∘ i₂ ] ∘ out Z ∘ h , i₂ ∘ i₁ ] 
      --   ∘ (idC +₁ f (! (algebras Y))) ∘ [ [ i₁ , i₂ ∘ i₂ ] ∘ out Y ∘ g , i₂ ∘ i₁ ] ∘ out X ≈˘⟨ refl⟩∘⟨ (refl⟩∘⟨ (refl⟩∘⟨ identityˡ)) ⟩ 
      --   out⁻¹ Z ∘ (idC +₁ f (! (algebras Z))) ∘ [ [ i₁ , i₂ ∘ i₂ ] ∘ out Z ∘ h , i₂ ∘ i₁ ] ∘ idC 
      --   ∘ (idC +₁ f (! (algebras Y))) ∘ [ [ i₁ , i₂ ∘ i₂ ] ∘ out Y ∘ g , i₂ ∘ i₁ ] ∘ out X ≈˘⟨ pullʳ (pullʳ (pullʳ (pullˡ (_≅_.isoʳ (out-≅ Y))))) ⟩ 
      --   (out⁻¹ Z ∘ (idC +₁ f (! (algebras Z))) ∘ [ [ i₁ , i₂ ∘ i₂ ] ∘ out Z ∘ h , i₂ ∘ i₁ ] ∘ out Y) 
      --   ∘ (out⁻¹ Y ∘ (idC +₁ f (! (algebras Y))) ∘ [ [ i₁ , i₂ ∘ i₂ ] ∘ out Y ∘ g , i₂ ∘ i₁ ] ∘ out X) ≈˘⟨ (refl⟩∘⟨ (pullʳ inject₁)) ⟩∘⟨ (refl⟩∘⟨ (pullʳ inject₁)) ⟩ 
      --   (out⁻¹ Z ∘ ((idC +₁ (f (! (algebras Z)))) ∘ F-Coalgebra.α (alg h)) ∘ i₁) ∘ (out⁻¹ Y ∘ ((idC +₁ (f (! (algebras Y)))) ∘ F-Coalgebra.α (alg g)) ∘ i₁) ≈˘⟨ (refl⟩∘⟨ (pullˡ (commutes (! (algebras Z))))) ⟩∘⟨ refl⟩∘⟨ (pullˡ (commutes (! (algebras Y)))) ⟩ 
      --   (out⁻¹ Z ∘ out Z ∘ extend h) ∘ (out⁻¹ Y ∘ out Y ∘ extend g) ≈˘⟨ ((insertˡ (_≅_.isoˡ (out-≅ Z)))) ⟩∘⟨ ((insertˡ (_≅_.isoˡ (out-≅ Y)))) ⟩ 
      --   extend h ∘ extend g ∎


        -- begin 
        -- extend (extend h ∘ g) ≈⟨ (insertˡ (_≅_.isoˡ (out-≅ Z))) ⟩ 
        -- out⁻¹ Z ∘ out Z ∘ extend (extend h ∘ g) ≈⟨ refl⟩∘⟨ extendlaw (extend h ∘ g) ⟩ 
        -- out⁻¹ Z ∘ [ out Z ∘ extend h ∘ g , i₂ ∘ extend (extend h ∘ g) ] ∘ out X ≈⟨ {!   !} ⟩ 
        -- {!   !} ≈⟨ {!   !} ⟩ 
        -- {!   !} ≈⟨ {!   !} ⟩ 
        -- {!   !} ≈⟨ {!   !} ⟩ 
        -- {!   !} ≈˘⟨ {!   !} ⟩ 
        -- {!   !} ≈˘⟨ {!   !} ⟩ 
        -- out⁻¹ Z ∘ [ [ out Z ∘ h , i₂ ∘ extend h ] ∘ out Y ∘ g , (i₂ ∘ extend h) ∘ extend g ] ∘ out X ≈˘⟨ refl⟩∘⟨ (([]-cong₂ refl (pullˡ inject₂)) ⟩∘⟨refl) ⟩ 
        -- out⁻¹ Z ∘ [ [ out Z ∘ h , i₂ ∘ extend h ] ∘ out Y ∘ g , [ out Z ∘ h , i₂ ∘ extend h ] ∘ i₂ ∘ extend g ] ∘ out X ≈˘⟨ refl⟩∘⟨ (pullˡ ∘[]) ⟩ 
        -- out⁻¹ Z ∘ [ out Z ∘ h , i₂ ∘ extend h ] ∘ [ out Y ∘ g , i₂ ∘ extend g ] ∘ out X ≈˘⟨ refl⟩∘⟨ (refl⟩∘⟨ identityˡ) ⟩ 
        -- out⁻¹ Z ∘ [ out Z ∘ h , i₂ ∘ extend h ] ∘ idC ∘ [ out Y ∘ g , i₂ ∘ extend g ] ∘ out X ≈˘⟨ pullʳ (pullʳ (pullˡ (_≅_.isoʳ (out-≅ Y)))) ⟩ 
        -- (out⁻¹ Z ∘ [ out Z ∘ h , i₂ ∘ extend h ] ∘ out Y) ∘ out⁻¹ Y ∘ [ out Y ∘ g , i₂ ∘ extend g ] ∘ out X ≈˘⟨ (refl⟩∘⟨ extendlaw h) ⟩∘⟨ (refl⟩∘⟨ extendlaw g) ⟩ 
        -- (out⁻¹ Z ∘ out Z ∘ extend h) ∘ (out⁻¹ Y ∘ out Y ∘ extend g) ≈˘⟨ ((insertˡ (_≅_.isoˡ (out-≅ Z)))) ⟩∘⟨ ((insertˡ (_≅_.isoˡ (out-≅ Y)))) ⟩ 
        -- extend h ∘ extend g ∎


        --begin 
        -- f (! (algebras Z) {A = alg (extend h ∘ g)}) ∘ i₁ {A = D₀ X} {B = D₀ Z} ≈⟨ (!-unique (algebras Z) (record { f = {! (f (! (algebras Z) {A = alg h}) ∘ i₁) ∘ f (! (algebras Y) {A = alg g})  !} ; commutes = {!   !} })) ⟩∘⟨refl ⟩ 
        -- {!   !} ∘ i₁ ≈⟨ {!   !} ⟩
        -- (f (! (algebras Z) {A = alg h}) ∘ i₁) ∘ f (! (algebras Y) {A = alg g}) ∘ i₁ ∎
      ; sym-assoc = λ {X} {Y} {Z} {g} {h} → {!   !}
      ; extend-≈ = λ {X} {Y} {f} {g} eq → begin 
          F-Coalgebra-Morphism.f (Terminal.! (algebras Y) {A = alg f }) ∘ i₁ {B = D₀ Y} ≈⟨ (Terminal.!-unique (algebras Y) (record { f = (F-Coalgebra-Morphism.f (Terminal.! (algebras Y) {A = alg g }) ∘ idC) ; commutes = begin 
            F-Coalgebra.α (Terminal.⊤ (algebras Y)) ∘ F-Coalgebra-Morphism.f (Terminal.! (algebras Y)) ∘ idC ≈⟨ refl⟩∘⟨ identityʳ ⟩ 
            F-Coalgebra.α (Terminal.⊤ (algebras Y)) ∘ F-Coalgebra-Morphism.f (Terminal.! (algebras Y)) ≈⟨ F-Coalgebra-Morphism.commutes (Terminal.! (algebras Y)) ⟩ 
            Functor.F₁ (delayF Y) (F-Coalgebra-Morphism.f (Terminal.! (algebras Y))) ∘ F-Coalgebra.α (alg g) ≈˘⟨ (Functor.F-resp-≈ (delayF Y) identityʳ) ⟩∘⟨ (αf≈αg eq) ⟩ 
            Functor.F₁ (delayF Y) (F-Coalgebra-Morphism.f (Terminal.! (algebras Y)) ∘ idC) ∘ F-Coalgebra.α (alg f) ∎ })) ⟩∘⟨refl ⟩ 
          (F-Coalgebra-Morphism.f (Terminal.! (algebras Y) {A = alg g }) ∘ idC) ∘ i₁ {B = D₀ Y} ≈⟨ identityʳ ⟩∘⟨refl ⟩
          extend g ∎
      })
      where
        open Terminal
        alg' : ∀ {X Y} → F-Coalgebra (delayF Y)
        alg' {X} {Y} = record { A = D₀ X ; α = i₂ }
        module _ {X Y : Obj} (f : X ⇒ D₀ Y) where
          -- open Terminal (algebras Y) using (!; ⊤-id)
          alg : F-Coalgebra (delayF Y)
          alg = record { A = D₀ X + D₀ Y ; α = [ [ [ i₁ , i₂ ∘ i₂ ] ∘ (out Y ∘ f) , i₂ ∘ i₁ ] ∘ out X , (idC +₁ i₂) ∘ out Y ] }  -- (idC +₁ (idC +₁ [ idC , idC ]) ∘ _≅_.to +-assoc ∘ _≅_.to +-comm)
          extend : D₀ X ⇒ D₀ Y
          extend = F-Coalgebra-Morphism.f (! (algebras Y) {A = alg}) ∘ i₁ {B = D₀ Y}
          !∘i₂ : F-Coalgebra-Morphism.f (! (algebras Y) {A = alg}) ∘ i₂ ≈ idC
          !∘i₂ = ⊤-id (algebras Y) (F-Coalgebras (delayF Y) [ ! (algebras Y) ∘ record { f = i₂ ; commutes = inject₂ } ] )
          extendlaw : out Y ∘ extend ≈ [ out Y ∘ f , i₂ ∘ extend ] ∘ out X
          extendlaw = begin 
            out Y ∘ extend ≈⟨ pullˡ (F-Coalgebra-Morphism.commutes (! (algebras Y) {A = alg})) ⟩ 
            ((idC +₁ (F-Coalgebra-Morphism.f (! (algebras Y)))) ∘ F-Coalgebra.α alg) ∘ coproduct.i₁                   ≈⟨ pullʳ inject₁ ⟩
            (idC +₁ (F-Coalgebra-Morphism.f (! (algebras Y)))) ∘ [ [ i₁ , i₂ ∘ i₂ ] ∘ (out Y ∘ f) , i₂ ∘ i₁ ] ∘ out X ≈⟨ pullˡ ∘[] ⟩
            [ (idC +₁ (F-Coalgebra-Morphism.f (! (algebras Y)))) ∘ [ i₁ , i₂ ∘ i₂ ] ∘ (out Y ∘ f) 
            , (idC +₁ (F-Coalgebra-Morphism.f (! (algebras Y)))) ∘ i₂ ∘ i₁ ] ∘ out X                                  ≈⟨ ([]-cong₂ (pullˡ ∘[]) (pullˡ +₁∘i₂)) ⟩∘⟨refl ⟩
            [ [ (idC +₁ (F-Coalgebra-Morphism.f (! (algebras Y)))) ∘ i₁ 
              , (idC +₁ (F-Coalgebra-Morphism.f (! (algebras Y)))) ∘ i₂ ∘ i₂ ] ∘ (out Y ∘ f) 
            , (i₂ ∘ (F-Coalgebra-Morphism.f (! (algebras Y)))) ∘ i₁ ] ∘ out X                                         ≈⟨ ([]-cong₂ (([]-cong₂ +₁∘i₁ (pullˡ +₁∘i₂)) ⟩∘⟨refl) refl) ⟩∘⟨refl ⟩
            [ [ i₁ ∘ idC , (i₂ ∘ (F-Coalgebra-Morphism.f (! (algebras Y)))) ∘ i₂ ] ∘ (out Y ∘ f) 
            , (i₂ ∘ (F-Coalgebra-Morphism.f (! (algebras Y)))) ∘ i₁ ] ∘ out X                                         ≈⟨ ([]-cong₂ (elimˡ (([]-cong₂ identityʳ (cancelʳ !∘i₂)) ○ +-η)) assoc) ⟩∘⟨refl ⟩
            [ out Y ∘ f , i₂ ∘ extend ] ∘ out X ∎ 
          extend-unique : (g : D₀ X ⇒ D₀ Y) → extend ≈ g
          extend-unique g = {!   !}
            -- begin 
          --   F-Coalgebra-Morphism.f (! (algebras Y) {A = alg}) ∘ i₁ {B = D₀ Y} ≈⟨ (!-unique (algebras Y) (record { f = [ g , idC ] ; commutes = begin 
          --     out Y ∘ [ g , idC ] ≈⟨ ∘[] ⟩ 
          --     [ out Y ∘ g , out Y ∘ idC ] ≈⟨ []-cong₂ {!   !} identityʳ ⟩
          --     {!   !} ≈˘⟨ {!   !} ⟩
          --     [ ([ out Y , i₂ ] ∘ (f +₁ g)) ∘ out X , out Y ] ≈˘⟨ []-cong₂ (sym []∘+₁ ⟩∘⟨refl) refl ⟩
          --     [ [ out Y ∘ f , i₂ ∘ g ] ∘ out X , out Y ] ≈˘⟨ {!   !} ⟩
          --     [ [ [ i₁ , i₂ ∘ idC ] ∘ (out Y ∘ f) , i₂ ∘ g ] ∘ out X , out Y ] ≈˘⟨ []-cong₂ (([]-cong₂ (([]-cong₂ identityʳ (pullʳ inject₂)) ⟩∘⟨refl) refl) ⟩∘⟨refl) refl ⟩
          --     [ [ [ i₁ ∘ idC , (i₂ ∘ [ g , idC ]) ∘ i₂ ] ∘ (out Y ∘ f) , i₂ ∘ g ] ∘ out X , out Y ] ≈˘⟨ []-cong₂ (([]-cong₂ (([]-cong₂ +₁∘i₁ (pullˡ +₁∘i₂)) ⟩∘⟨refl) (pullʳ inject₁)) ⟩∘⟨refl) (elimˡ (Functor.identity (delayF Y))) ⟩
          --     [ [ [ (idC +₁ [ g , idC ]) ∘ i₁ , (idC +₁ [ g , idC ]) ∘ i₂ ∘ i₂ ] ∘ (out Y ∘ f) , (i₂ ∘ [ g , idC ]) ∘ i₁ ] ∘ out X , (idC +₁ idC) ∘ out Y ] ≈˘⟨ []-cong₂ (([]-cong₂ (pullˡ ∘[]) (pullˡ +₁∘i₂)) ⟩∘⟨refl) ((+₁-cong₂ identity² inject₂) ⟩∘⟨refl) ⟩
          --     [ [ (idC +₁ [ g , idC ]) ∘ [ i₁ , i₂ ∘ i₂ ] ∘ (out Y ∘ f) , (idC +₁ [ g , idC ]) ∘ i₂ ∘ i₁ ] ∘ out X , (idC ∘ idC +₁ [ g , idC ] ∘ i₂) ∘ out Y ] ≈˘⟨ []-cong₂ (pullˡ ∘[]) (pullˡ +₁∘+₁) ⟩
          --     [ (idC +₁ [ g , idC ]) ∘ [ [ i₁ , i₂ ∘ i₂ ] ∘ (out Y ∘ f) , i₂ ∘ i₁ ] ∘ out X , (idC +₁ [ g , idC ]) ∘ (idC +₁ i₂) ∘ out Y ] ≈˘⟨ ∘[] ⟩
          --     (idC +₁ [ g , idC ]) ∘ F-Coalgebra.α alg ∎ })) ⟩∘⟨refl ⟩ 
          --   [ g , idC ] ∘ i₁ ≈⟨ inject₁ ⟩
          --   g ∎
        αf≈αg : ∀ {X Y} {f g : X ⇒ D₀ Y} → f ≈ g → F-Coalgebra.α (alg f) ≈ F-Coalgebra.α (alg g)
        αf≈αg {X} {Y} {f} {g} eq = []-cong₂ ([]-cong₂ (refl⟩∘⟨ refl⟩∘⟨ eq) refl ⟩∘⟨refl) refl
        alg-f≈alg-g : ∀ {X Y} {f g : X ⇒ D₀ Y} → f ≈ g → M._≅_ (F-Coalgebras (delayF Y)) (alg f) (alg g)
        alg-f≈alg-g {X} {Y} {f} {g} eq = record 
          { from = record { f = idC ; commutes = begin 
            F-Coalgebra.α (alg g) ∘ idC ≈⟨ identityʳ ⟩ 
            F-Coalgebra.α (alg g) ≈⟨ ⟺ (αf≈αg eq) ⟩ 
            F-Coalgebra.α (alg f) ≈˘⟨ elimˡ (Functor.identity (delayF Y)) ⟩ 
            Functor.F₁ (delayF Y) idC ∘ F-Coalgebra.α (alg f) ∎ } 
          ; to = record { f = idC ; commutes = begin 
            F-Coalgebra.α (alg f) ∘ idC ≈⟨ identityʳ ⟩ 
            F-Coalgebra.α (alg f) ≈⟨ αf≈αg eq ⟩ 
            F-Coalgebra.α (alg g) ≈˘⟨ elimˡ (Functor.identity (delayF Y)) ⟩ 
            Functor.F₁ (delayF Y) idC ∘ F-Coalgebra.α (alg g) ∎ } 
          ; iso = record 
            { isoˡ = identity² 
            ; isoʳ = identity² 
            } 
          }
```

### Definition 30: Search-Algebras

TODO

### Proposition 31 : the category of uniform-iteration algebras coincides with the category of search-algebras

TODO