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
-
-module _ {ℓ : Level} {ℂ : Category ℓ ℓ} where
-  private
-    𝓢 = Sets ℓ
-    open Fun (opposite ℂ) 𝓢
-    prshf = presheaf ℂ
-    module ℂ = Category ℂ
+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.
-    _⇑_ = CatExponential.object
+-- 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
+    presheaf = Cat.Categories.Sets.presheaf ℂ
+    module ℂ = Category ℂ
 
     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