Formatting in yoneda

src/Cat/Category/Yoneda.agda

@@ -12,51 +12,52 @@ open import Cat.Category.Functor
 open import Cat.Equality
 
 open import Cat.Categories.Fun
-open import Cat.Categories.Sets
-open import Cat.Categories.Cat
+open import Cat.Categories.Sets hiding (presheaf)
+
+-- There is no (small) category of categories. So we won't use _⇑_ from
+-- `HasExponential`
+--
+--     open HasExponentials (Cat.hasExponentials ℓ unprovable) using (_⇑_)
+--
+-- In stead we'll use an ad-hoc definition -- which is definitionally equivalent
+-- to that other one - even without mentioning the category of categories.
+_⇑_ : {ℓ : Level} → Category ℓ ℓ → Category ℓ ℓ → Category ℓ ℓ
+_⇑_ = Fun.Fun
 
 module _ {ℓ : Level} {ℂ : Category ℓ ℓ} where
   private
     𝓢 = Sets ℓ
     open Fun (opposite ℂ) 𝓢
-    prshf = presheaf ℂ
+    presheaf = Cat.Categories.Sets.presheaf ℂ
     module ℂ = Category ℂ
 
-    -- There is no (small) category of categories. So we won't use _⇑_ from
-    -- `HasExponential`
-    --
-    --     open HasExponentials (Cat.hasExponentials ℓ unprovable) using (_⇑_)
-    --
-    -- In stead we'll use an ad-hoc definition -- which is definitionally
-    -- equivalent to that other one.
-    _⇑_ = CatExponential.object
-
     module _ {A B : ℂ.Object} (f : ℂ [ A , B ]) where
-      fmap : Transformation (prshf A) (prshf B)
+      fmap : Transformation (presheaf A) (presheaf B)
       fmap C x = ℂ [ f ∘ x ]
 
-      fmapNatural : Natural (prshf A) (prshf B) fmap
+      fmapNatural : Natural (presheaf A) (presheaf B) fmap
       fmapNatural g = funExt λ _ → ℂ.isAssociative
 
-      fmapNT : NaturalTransformation (prshf A) (prshf B)
+      fmapNT : NaturalTransformation (presheaf A) (presheaf B)
       fmapNT = fmap , fmapNatural
 
     rawYoneda : RawFunctor ℂ Fun
-    RawFunctor.omap rawYoneda = prshf
+    RawFunctor.omap rawYoneda = presheaf
     RawFunctor.fmap rawYoneda = fmapNT
+
     open RawFunctor rawYoneda hiding (fmap)
 
     isIdentity : IsIdentity
-    isIdentity {c} = lemSig (naturalIsProp {F = prshf c} {prshf c}) _ _ eq
+    isIdentity {c} = lemSig (naturalIsProp {F = presheaf c} {presheaf c}) _ _ eq
       where
-      eq : (λ C x → ℂ [ ℂ.𝟙 ∘ x ]) ≡ identityTrans (prshf c)
+      eq : (λ C x → ℂ [ ℂ.𝟙 ∘ x ]) ≡ identityTrans (presheaf c)
       eq = funExt λ A → funExt λ B → proj₂ ℂ.isIdentity
 
     isDistributive : IsDistributive
     isDistributive {A} {B} {C} {f = f} {g}
-      = lemSig (propIsNatural (prshf A) (prshf C)) _ _ eq
+      = lemSig (propIsNatural (presheaf A) (presheaf C)) _ _ eq
       where
-      T[_∘_]' = T[_∘_] {F = prshf A} {prshf B} {prshf C}
+      T[_∘_]' = T[_∘_] {F = presheaf A} {presheaf B} {presheaf C}
       eqq : (X : ℂ.Object) → (x : ℂ [ X , A ])
         → fmap (ℂ [ g ∘ f ]) X x ≡ T[ fmap g ∘ fmap f ]' X x
       eqq X x = begin
@@ -76,5 +77,5 @@ module _ {ℓ : Level} {ℂ : Category ℓ ℓ} where
       IsFunctor.isDistributive isFunctor = isDistributive
 
   yoneda : Functor ℂ Fun
-  Functor.raw yoneda = rawYoneda
+  Functor.raw       yoneda = rawYoneda
   Functor.isFunctor yoneda = isFunctor