Use new syntax in cat
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@ -18,7 +18,7 @@ open Equality.Data.Product
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open Functor
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open IsFunctor
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open Category hiding (_∘_)
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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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@ -45,7 +45,7 @@ module _ (ℓ ℓ' : Level) where
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]
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assc : H ∘f (G ∘f F) ≡ (H ∘f G) ∘f F
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assc = Functor≡ eq* eq→ (IsFunctor≡ eqI eqD)
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assc = Functor≡ eq* eq→
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module _ {ℂ 𝔻 : Category ℓ ℓ'} {F : Functor ℂ 𝔻} where
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module _ where
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@ -55,7 +55,7 @@ module _ (ℓ ℓ' : Level) where
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-- lemmm : func→ {C = A} {D = B} (f ∘f identity) ≡ func→ f
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eq→ : PathP
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(λ i →
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{x y : Object ℂ} → Arrow ℂ x y → Arrow 𝔻 (func* F x) (func* F y))
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{x y : Object ℂ} → ℂ [ x , y ] → 𝔻 [ func* F x , func* F y ])
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(func→ (F ∘f identity)) (func→ F)
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eq→ = refl
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postulate
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@ -69,14 +69,14 @@ module _ (ℓ ℓ' : Level) where
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eq→ i (ℂ [ g ∘ f ]) ≡ 𝔻 [ eq→ i g ∘ eq→ i f ])
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((F ∘f identity) .isFunctor .distrib) (F .isFunctor .distrib)
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ident-r : F ∘f identity ≡ F
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ident-r = Functor≡ eq* eq→ (IsFunctor≡ eqI-r eqD-r)
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ident-r = Functor≡ eq* eq→
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module _ where
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private
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postulate
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eq* : (identity ∘f F) .func* ≡ F .func*
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eq* : func* (identity ∘f F) ≡ func* F
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eq→ : PathP
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(λ i → {x y : Object ℂ} → ℂ [ x , y ] → 𝔻 [ eq* i x , eq* i y ])
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((identity ∘f F) .func→) (F .func→)
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(func→ (identity ∘f F)) (func→ F)
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eqI : (λ i → ∀ {A : Object ℂ} → eq→ i (𝟙 ℂ {A}) ≡ 𝟙 𝔻 {eq* i A})
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[ ((identity ∘f F) .isFunctor .ident) ≡ (F .isFunctor .ident) ]
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eqD : PathP (λ i → {A B C : Object ℂ} {f : ℂ [ A , B ]} {g : ℂ [ B , C ]}
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@ -84,7 +84,7 @@ module _ (ℓ ℓ' : Level) where
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((identity ∘f F) .isFunctor .distrib) (F .isFunctor .distrib)
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-- (λ z → eq* i z) (eq→ i)
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ident-l : identity ∘f F ≡ F
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ident-l = Functor≡ eq* eq→ λ i → record { ident = eqI i ; distrib = eqD i }
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ident-l = Functor≡ eq* eq→
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RawCat : RawCategory (lsuc (ℓ ⊔ ℓ')) (ℓ ⊔ ℓ')
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RawCat =
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@ -104,11 +104,11 @@ module _ (ℓ ℓ' : Level) where
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:isCategory: : IsCategory RawCat
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assoc :isCategory: {f = F} {G} {H} = assc {F = F} {G = G} {H = H}
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ident :isCategory: = ident-r , ident-l
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arrow-is-set :isCategory: = {!!}
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arrowIsSet :isCategory: = {!!}
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univalent :isCategory: = {!!}
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Cat : Category (lsuc (ℓ ⊔ ℓ')) (ℓ ⊔ ℓ')
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raw Cat = RawCat
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Category.raw Cat = RawCat
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module _ {ℓ ℓ' : Level} where
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module _ (ℂ 𝔻 : Category ℓ ℓ') where
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@ -116,7 +116,7 @@ module _ {ℓ ℓ' : Level} where
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Catt = Cat ℓ ℓ'
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:Object: = Object ℂ × Object 𝔻
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:Arrow: : :Object: → :Object: → Set ℓ'
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:Arrow: (c , d) (c' , d') = Arrow ℂ c c' × Arrow 𝔻 d d'
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:Arrow: (c , d) (c' , d') = ℂ [ c , c' ] × 𝔻 [ d , d' ]
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:𝟙: : {o : :Object:} → :Arrow: o o
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:𝟙: = 𝟙 ℂ , 𝟙 𝔻
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_:⊕:_ :
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@ -132,8 +132,8 @@ module _ {ℓ ℓ' : Level} where
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RawCategory.𝟙 :rawProduct: = :𝟙:
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RawCategory._∘_ :rawProduct: = _:⊕:_
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module C = IsCategory (ℂ .isCategory)
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module D = IsCategory (𝔻 .isCategory)
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module C = Category ℂ
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module D = Category 𝔻
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postulate
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issSet : {A B : RawCategory.Object :rawProduct:} → isSet (RawCategory.Arrow :rawProduct: A B)
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instance
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@ -150,17 +150,23 @@ module _ {ℓ ℓ' : Level} where
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IsCategory.ident :isCategory:
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= Σ≡ (fst C.ident) (fst D.ident)
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, Σ≡ (snd C.ident) (snd D.ident)
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IsCategory.arrow-is-set :isCategory: = issSet
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IsCategory.arrowIsSet :isCategory: = issSet
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IsCategory.univalent :isCategory: = {!!}
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:product: : Category ℓ ℓ'
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raw :product: = :rawProduct:
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Category.raw :product: = :rawProduct:
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proj₁ : Catt [ :product: , ℂ ]
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proj₁ = record { func* = fst ; func→ = fst ; isFunctor = record { ident = refl ; distrib = refl } }
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proj₁ = record
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{ raw = record { func* = fst ; func→ = fst }
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; isFunctor = record { ident = refl ; distrib = refl }
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}
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proj₂ : Catt [ :product: , 𝔻 ]
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proj₂ = record { func* = snd ; func→ = snd ; isFunctor = record { ident = refl ; distrib = refl } }
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proj₂ = record
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{ raw = record { func* = snd ; func→ = snd }
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; isFunctor = record { ident = refl ; distrib = refl }
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}
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module _ {X : Object Catt} (x₁ : Catt [ X , ℂ ]) (x₂ : Catt [ X , 𝔻 ]) where
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open Functor
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@ -232,7 +238,7 @@ module _ (ℓ : Level) where
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:obj: = Fun {ℂ = ℂ} {𝔻 = 𝔻}
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:func*: : Functor ℂ 𝔻 × Object ℂ → Object 𝔻
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:func*: (F , A) = F .func* A
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:func*: (F , A) = func* F A
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module _ {dom cod : Functor ℂ 𝔻 × Object ℂ} where
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private
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@ -247,27 +253,27 @@ module _ (ℓ : Level) where
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B = proj₂ cod
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:func→: : (pobj : NaturalTransformation F G × ℂ [ A , B ])
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→ 𝔻 [ F .func* A , G .func* B ]
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→ 𝔻 [ func* F A , func* G B ]
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:func→: ((θ , θNat) , f) = result
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where
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θA : 𝔻 [ F .func* A , G .func* A ]
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θA : 𝔻 [ func* F A , func* G A ]
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θA = θ A
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θB : 𝔻 [ F .func* B , G .func* B ]
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θB : 𝔻 [ func* F B , func* G B ]
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θB = θ B
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F→f : 𝔻 [ F .func* A , F .func* B ]
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F→f = F .func→ f
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G→f : 𝔻 [ G .func* A , G .func* B ]
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G→f = G .func→ f
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l : 𝔻 [ F .func* A , G .func* B ]
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F→f : 𝔻 [ func* F A , func* F B ]
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F→f = func→ F f
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G→f : 𝔻 [ func* G A , func* G B ]
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G→f = func→ G f
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l : 𝔻 [ func* F A , func* G B ]
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l = 𝔻 [ θB ∘ F→f ]
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r : 𝔻 [ F .func* A , G .func* B ]
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r : 𝔻 [ func* F A , func* G B ]
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r = 𝔻 [ G→f ∘ θA ]
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-- There are two choices at this point,
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-- but I suppose the whole point is that
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-- by `θNat f` we have `l ≡ r`
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-- lem : 𝔻 [ θ B ∘ F .func→ f ] ≡ 𝔻 [ G .func→ f ∘ θ A ]
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-- lem = θNat f
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result : 𝔻 [ F .func* A , G .func* B ]
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result : 𝔻 [ func* F A , func* G B ]
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result = l
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_×p_ = product
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@ -285,16 +291,16 @@ module _ (ℓ : Level) where
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-- where
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-- open module 𝔻 = IsCategory (𝔻 .isCategory)
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-- Unfortunately the equational version has some ambigous arguments.
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:ident: : :func→: {c} {c} (identityNat F , 𝟙 ℂ {o = proj₂ c}) ≡ 𝟙 𝔻
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:ident: : :func→: {c} {c} (identityNat F , 𝟙 ℂ {A = proj₂ c}) ≡ 𝟙 𝔻
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:ident: = begin
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:func→: {c} {c} (𝟙 (Product.obj (:obj: ×p ℂ)) {c}) ≡⟨⟩
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:func→: {c} {c} (identityNat F , 𝟙 ℂ) ≡⟨⟩
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𝔻 [ identityTrans F C ∘ F .func→ (𝟙 ℂ)] ≡⟨⟩
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𝔻 [ 𝟙 𝔻 ∘ F .func→ (𝟙 ℂ)] ≡⟨ proj₂ 𝔻.ident ⟩
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F .func→ (𝟙 ℂ) ≡⟨ F.ident ⟩
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𝔻 [ identityTrans F C ∘ func→ F (𝟙 ℂ)] ≡⟨⟩
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𝔻 [ 𝟙 𝔻 ∘ func→ F (𝟙 ℂ)] ≡⟨ proj₂ 𝔻.ident ⟩
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func→ F (𝟙 ℂ) ≡⟨ F.ident ⟩
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𝟙 𝔻 ∎
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where
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open module 𝔻 = IsCategory (𝔻 .isCategory)
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open module 𝔻 = Category 𝔻
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open module F = IsFunctor (F .isFunctor)
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module _ {F×A G×B H×C : Functor ℂ 𝔻 × Object ℂ} where
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@ -331,35 +337,38 @@ module _ (ℓ : Level) where
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ηθNat = proj₂ ηθNT
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:distrib: :
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𝔻 [ 𝔻 [ η C ∘ θ C ] ∘ F .func→ ( ℂ [ g ∘ f ] ) ]
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≡ 𝔻 [ 𝔻 [ η C ∘ G .func→ g ] ∘ 𝔻 [ θ B ∘ F .func→ f ] ]
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𝔻 [ 𝔻 [ η C ∘ θ C ] ∘ func→ F ( ℂ [ g ∘ f ] ) ]
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≡ 𝔻 [ 𝔻 [ η C ∘ func→ G g ] ∘ 𝔻 [ θ B ∘ func→ F f ] ]
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:distrib: = begin
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𝔻 [ (ηθ C) ∘ F .func→ (ℂ [ g ∘ f ]) ]
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𝔻 [ (ηθ C) ∘ func→ F (ℂ [ g ∘ f ]) ]
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≡⟨ ηθNat (ℂ [ g ∘ f ]) ⟩
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𝔻 [ H .func→ (ℂ [ g ∘ f ]) ∘ (ηθ A) ]
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𝔻 [ func→ H (ℂ [ g ∘ f ]) ∘ (ηθ A) ]
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≡⟨ cong (λ φ → 𝔻 [ φ ∘ ηθ A ]) (H.distrib) ⟩
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𝔻 [ 𝔻 [ H .func→ g ∘ H .func→ f ] ∘ (ηθ A) ]
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𝔻 [ 𝔻 [ func→ H g ∘ func→ H f ] ∘ (ηθ A) ]
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≡⟨ sym assoc ⟩
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𝔻 [ H .func→ g ∘ 𝔻 [ H .func→ f ∘ ηθ A ] ]
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≡⟨ cong (λ φ → 𝔻 [ H .func→ g ∘ φ ]) assoc ⟩
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𝔻 [ H .func→ g ∘ 𝔻 [ 𝔻 [ H .func→ f ∘ η A ] ∘ θ A ] ]
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≡⟨ cong (λ φ → 𝔻 [ H .func→ g ∘ φ ]) (cong (λ φ → 𝔻 [ φ ∘ θ A ]) (sym (ηNat f))) ⟩
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𝔻 [ H .func→ g ∘ 𝔻 [ 𝔻 [ η B ∘ G .func→ f ] ∘ θ A ] ]
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≡⟨ cong (λ φ → 𝔻 [ H .func→ g ∘ φ ]) (sym assoc) ⟩
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𝔻 [ H .func→ g ∘ 𝔻 [ η B ∘ 𝔻 [ G .func→ f ∘ θ A ] ] ] ≡⟨ assoc ⟩
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𝔻 [ 𝔻 [ H .func→ g ∘ η B ] ∘ 𝔻 [ G .func→ f ∘ θ A ] ]
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≡⟨ cong (λ φ → 𝔻 [ φ ∘ 𝔻 [ G .func→ f ∘ θ A ] ]) (sym (ηNat g)) ⟩
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𝔻 [ 𝔻 [ η C ∘ G .func→ g ] ∘ 𝔻 [ G .func→ f ∘ θ A ] ]
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≡⟨ cong (λ φ → 𝔻 [ 𝔻 [ η C ∘ G .func→ g ] ∘ φ ]) (sym (θNat f)) ⟩
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𝔻 [ 𝔻 [ η C ∘ G .func→ g ] ∘ 𝔻 [ θ B ∘ F .func→ f ] ] ∎
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𝔻 [ func→ H g ∘ 𝔻 [ func→ H f ∘ ηθ A ] ]
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≡⟨ cong (λ φ → 𝔻 [ func→ H g ∘ φ ]) assoc ⟩
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𝔻 [ func→ H g ∘ 𝔻 [ 𝔻 [ func→ H f ∘ η A ] ∘ θ A ] ]
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≡⟨ cong (λ φ → 𝔻 [ func→ H g ∘ φ ]) (cong (λ φ → 𝔻 [ φ ∘ θ A ]) (sym (ηNat f))) ⟩
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𝔻 [ func→ H g ∘ 𝔻 [ 𝔻 [ η B ∘ func→ G f ] ∘ θ A ] ]
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≡⟨ cong (λ φ → 𝔻 [ func→ H g ∘ φ ]) (sym assoc) ⟩
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𝔻 [ func→ H g ∘ 𝔻 [ η B ∘ 𝔻 [ func→ G f ∘ θ A ] ] ]
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≡⟨ assoc ⟩
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𝔻 [ 𝔻 [ func→ H g ∘ η B ] ∘ 𝔻 [ func→ G f ∘ θ A ] ]
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≡⟨ cong (λ φ → 𝔻 [ φ ∘ 𝔻 [ func→ G f ∘ θ A ] ]) (sym (ηNat g)) ⟩
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𝔻 [ 𝔻 [ η C ∘ func→ G g ] ∘ 𝔻 [ func→ G f ∘ θ A ] ]
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≡⟨ cong (λ φ → 𝔻 [ 𝔻 [ η C ∘ func→ G g ] ∘ φ ]) (sym (θNat f)) ⟩
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𝔻 [ 𝔻 [ η C ∘ func→ G g ] ∘ 𝔻 [ θ B ∘ func→ F f ] ] ∎
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where
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open IsCategory (𝔻 .isCategory)
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open module H = IsFunctor (H .isFunctor)
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open Category 𝔻
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module H = IsFunctor (H .isFunctor)
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:eval: : Functor ((:obj: ×p ℂ) .Product.obj) 𝔻
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:eval: = record
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{ raw = record
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{ func* = :func*:
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; func→ = λ {dom} {cod} → :func→: {dom} {cod}
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}
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; isFunctor = record
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{ ident = λ {o} → :ident: {o}
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; distrib = λ {f u n k y} → :distrib: {f} {u} {n} {k} {y}
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@ -371,7 +380,7 @@ module _ (ℓ : Level) where
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postulate
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transpose : Functor 𝔸 :obj:
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eq : Catℓ [ :eval: ∘ (parallelProduct transpose (𝟙 Catℓ {o = ℂ})) ] ≡ F
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eq : Catℓ [ :eval: ∘ (parallelProduct transpose (𝟙 Catℓ {A = ℂ})) ] ≡ F
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-- eq : Catℓ [ :eval: ∘ (HasProducts._|×|_ hasProducts transpose (𝟙 Catℓ {o = ℂ})) ] ≡ F
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-- eq' : (Catℓ [ :eval: ∘
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-- (record { product = product } HasProducts.|×| transpose)
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