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@ -18,8 +18,6 @@ open import Cat.Category.NaturalTransformation
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open import Cat.Equality
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open Equality.Data.Product
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open Category using (Object ; 𝟙)
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-- The category of categories
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module _ (ℓ ℓ' : Level) where
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private
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@ -35,19 +33,18 @@ module _ (ℓ ℓ' : Level) where
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ident-l = Functor≡ refl
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RawCat : RawCategory (lsuc (ℓ ⊔ ℓ')) (ℓ ⊔ ℓ')
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RawCat =
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record
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{ Object = Category ℓ ℓ'
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; Arrow = Functor
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; 𝟙 = identity
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; _∘_ = F[_∘_]
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}
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RawCategory.Object RawCat = Category ℓ ℓ'
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RawCategory.Arrow RawCat = Functor
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RawCategory.𝟙 RawCat = identity
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RawCategory._∘_ RawCat = F[_∘_]
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private
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open RawCategory RawCat
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isAssociative : IsAssociative
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isAssociative {f = F} {G} {H} = assc {F = F} {G = G} {H = H}
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ident : IsIdentity identity
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ident = ident-l , ident-r
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isIdentity : IsIdentity identity
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isIdentity = ident-l , ident-r
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-- NB! `ArrowsAreSets RawCat` is *not* provable. The type of functors,
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-- however, form a groupoid! Therefore there is no (1-)category of
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@ -55,7 +52,7 @@ module _ (ℓ ℓ' : Level) where
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--
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-- Because of this there is no category of categories.
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Cat : (unprovable : IsCategory RawCat) → Category (lsuc (ℓ ⊔ ℓ')) (ℓ ⊔ ℓ')
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Category.raw (Cat _) = RawCat
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Category.raw (Cat _) = RawCat
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Category.isCategory (Cat unprovable) = unprovable
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-- | In the following we will pretend there is a category of categories when
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@ -72,28 +69,31 @@ module CatProduct {ℓ ℓ' : Level} (ℂ 𝔻 : Category ℓ ℓ') where
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module ℂ = Category ℂ
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module 𝔻 = Category 𝔻
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Obj = Object ℂ × Object 𝔻
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Arr : Obj → Obj → Set ℓ'
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Arr (c , d) (c' , d') = ℂ [ c , c' ] × 𝔻 [ d , d' ]
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𝟙' : {o : Obj} → Arr o o
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𝟙' = 𝟙 ℂ , 𝟙 𝔻
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_∘_ :
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{a b c : Obj} →
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Arr b c →
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Arr a b →
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Arr a c
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_∘_ = λ { (bc∈C , bc∈D) (ab∈C , ab∈D) → ℂ [ bc∈C ∘ ab∈C ] , 𝔻 [ bc∈D ∘ ab∈D ]}
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module _ where
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private
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Obj = ℂ.Object × 𝔻.Object
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Arr : Obj → Obj → Set ℓ'
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Arr (c , d) (c' , d') = ℂ [ c , c' ] × 𝔻 [ d , d' ]
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𝟙 : {o : Obj} → Arr o o
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𝟙 = ℂ.𝟙 , 𝔻.𝟙
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_∘_ :
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{a b c : Obj} →
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Arr b c →
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Arr a b →
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Arr a c
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_∘_ = λ { (bc∈C , bc∈D) (ab∈C , ab∈D) → ℂ [ bc∈C ∘ ab∈C ] , 𝔻 [ bc∈D ∘ ab∈D ]}
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rawProduct : RawCategory ℓ ℓ'
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RawCategory.Object rawProduct = Obj
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RawCategory.Arrow rawProduct = Arr
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RawCategory.𝟙 rawProduct = 𝟙'
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RawCategory._∘_ rawProduct = _∘_
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open RawCategory rawProduct
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rawProduct : RawCategory ℓ ℓ'
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RawCategory.Object rawProduct = Obj
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RawCategory.Arrow rawProduct = Arr
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RawCategory.𝟙 rawProduct = 𝟙
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RawCategory._∘_ rawProduct = _∘_
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open RawCategory rawProduct
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arrowsAreSets : ArrowsAreSets
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arrowsAreSets = setSig {sA = ℂ.arrowsAreSets} {sB = λ x → 𝔻.arrowsAreSets}
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isIdentity : IsIdentity 𝟙'
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isIdentity : IsIdentity 𝟙
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isIdentity
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= Σ≡ (fst ℂ.isIdentity) (fst 𝔻.isIdentity)
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, Σ≡ (snd ℂ.isIdentity) (snd 𝔻.isIdentity)
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@ -189,102 +189,65 @@ module CatExponential {ℓ : Level} (ℂ 𝔻 : Category ℓ ℓ) where
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Categoryℓ = Category ℓ ℓ
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open Fun ℂ 𝔻 renaming (identity to idN)
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omap : Functor ℂ 𝔻 × Object ℂ → Object 𝔻
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omap : Functor ℂ 𝔻 × ℂ.Object → 𝔻.Object
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omap (F , A) = Functor.omap F A
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-- The exponential object
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object : Categoryℓ
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object = Fun
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module _ {dom cod : Functor ℂ 𝔻 × Object ℂ} where
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module _ {dom cod : Functor ℂ 𝔻 × ℂ.Object} where
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open Σ dom renaming (proj₁ to F ; proj₂ to A)
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open Σ cod renaming (proj₁ to G ; proj₂ to B)
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private
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F : Functor ℂ 𝔻
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F = proj₁ dom
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A : Object ℂ
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A = proj₂ dom
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G : Functor ℂ 𝔻
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G = proj₁ cod
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B : Object ℂ
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B = proj₂ cod
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module F = Functor F
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module G = Functor G
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fmap : (pobj : NaturalTransformation F G × ℂ [ A , B ])
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→ 𝔻 [ F.omap A , G.omap B ]
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fmap ((θ , θNat) , f) = result
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where
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θA : 𝔻 [ F.omap A , G.omap A ]
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θA = θ A
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θB : 𝔻 [ F.omap B , G.omap B ]
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θB = θ B
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F→f : 𝔻 [ F.omap A , F.omap B ]
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F→f = F.fmap f
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G→f : 𝔻 [ G.omap A , G.omap B ]
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G→f = G.fmap f
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l : 𝔻 [ F.omap A , G.omap B ]
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l = 𝔻 [ θB ∘ F.fmap f ]
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r : 𝔻 [ F.omap A , G.omap B ]
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r = 𝔻 [ G.fmap f ∘ θA ]
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result : 𝔻 [ F.omap A , G.omap B ]
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result = l
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fmap ((θ , θNat) , f) = 𝔻 [ θ B ∘ F.fmap f ]
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-- Alternatively:
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--
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-- fmap ((θ , θNat) , f) = 𝔻 [ G.fmap f ∘ θ A ]
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--
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-- Since they are equal by naturality of θ.
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open CatProduct renaming (object to _⊗_) using ()
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module _ {c : Functor ℂ 𝔻 × Object ℂ} where
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private
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F : Functor ℂ 𝔻
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F = proj₁ c
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C : Object ℂ
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C = proj₂ c
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module _ {c : Functor ℂ 𝔻 × ℂ.Object} where
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open Σ c renaming (proj₁ to F ; proj₂ to C)
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ident : fmap {c} {c} (NT.identity F , 𝟙 ℂ {A = proj₂ c}) ≡ 𝟙 𝔻
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ident : fmap {c} {c} (NT.identity F , ℂ.𝟙 {A = proj₂ c}) ≡ 𝔻.𝟙
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ident = begin
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fmap {c} {c} (𝟙 (object ⊗ ℂ) {c}) ≡⟨⟩
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fmap {c} {c} (idN F , 𝟙 ℂ) ≡⟨⟩
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𝔻 [ identityTrans F C ∘ F.fmap (𝟙 ℂ)] ≡⟨⟩
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𝔻 [ 𝟙 𝔻 ∘ F.fmap (𝟙 ℂ)] ≡⟨ 𝔻.leftIdentity ⟩
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F.fmap (𝟙 ℂ) ≡⟨ F.isIdentity ⟩
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𝟙 𝔻 ∎
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fmap {c} {c} (Category.𝟙 (object ⊗ ℂ) {c}) ≡⟨⟩
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fmap {c} {c} (idN F , ℂ.𝟙) ≡⟨⟩
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𝔻 [ identityTrans F C ∘ F.fmap ℂ.𝟙 ] ≡⟨⟩
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𝔻 [ 𝔻.𝟙 ∘ F.fmap ℂ.𝟙 ] ≡⟨ 𝔻.leftIdentity ⟩
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F.fmap ℂ.𝟙 ≡⟨ F.isIdentity ⟩
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𝔻.𝟙 ∎
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where
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module F = Functor F
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module _ {F×A G×B H×C : Functor ℂ 𝔻 × Object ℂ} where
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module _ {F×A G×B H×C : Functor ℂ 𝔻 × ℂ.Object} where
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open Σ F×A renaming (proj₁ to F ; proj₂ to A)
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open Σ G×B renaming (proj₁ to G ; proj₂ to B)
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open Σ H×C renaming (proj₁ to H ; proj₂ to C)
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private
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F = F×A .proj₁
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A = F×A .proj₂
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G = G×B .proj₁
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B = G×B .proj₂
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H = H×C .proj₁
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C = H×C .proj₂
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module F = Functor F
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module G = Functor G
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module H = Functor H
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module _
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-- NaturalTransformation F G × ℂ .Arrow A B
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{θ×f : NaturalTransformation F G × ℂ [ A , B ]}
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{η×g : NaturalTransformation G H × ℂ [ B , C ]} where
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open Σ θ×f renaming (proj₁ to θNT ; proj₂ to f)
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open Σ θNT renaming (proj₁ to θ ; proj₂ to θNat)
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open Σ η×g renaming (proj₁ to ηNT ; proj₂ to g)
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open Σ ηNT renaming (proj₁ to η ; proj₂ to ηNat)
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private
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θ : Transformation F G
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θ = proj₁ (proj₁ θ×f)
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θNat : Natural F G θ
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θNat = proj₂ (proj₁ θ×f)
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f : ℂ [ A , B ]
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f = proj₂ θ×f
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η : Transformation G H
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η = proj₁ (proj₁ η×g)
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ηNat : Natural G H η
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ηNat = proj₂ (proj₁ η×g)
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g : ℂ [ B , C ]
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g = proj₂ η×g
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ηθNT : NaturalTransformation F H
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ηθNT = Category._∘_ Fun {F} {G} {H} (η , ηNat) (θ , θNat)
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ηθ = proj₁ ηθNT
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ηθNat = proj₂ ηθNT
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ηθNT = NT[_∘_] {F} {G} {H} ηNT θNT
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open Σ ηθNT renaming (proj₁ to ηθ ; proj₂ to ηθNat)
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isDistributive :
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𝔻 [ 𝔻 [ η C ∘ θ C ] ∘ F.fmap ( ℂ [ g ∘ f ] ) ]
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