Implement category of presheaves

src/Cat/Categories/Fun.agda

@@ -62,9 +62,7 @@ module Fun {ℓc ℓc' ℓd ℓd' : Level} (ℂ : Category ℓc ℓc') (𝔻 : C
     _h⊕g_ = NT[_∘_] {B} {C} {D}
     :isAssociative: : L ≡ R
     :isAssociative: = lemSig (naturalIsProp {F = A} {D})
-      L R (funExt (λ x → isAssociative))
-      where
-        open Category 𝔻
+      L R (funExt (λ x → 𝔻.isAssociative))
 
   private
     module _ {A B : Functor ℂ 𝔻} {f : NaturalTransformation A B} where
@@ -107,14 +105,22 @@ module Fun {ℓc ℓc' ℓd ℓd' : Level} (ℂ : Category ℓc ℓc') (𝔻 : C
   Category.raw Fun = RawFun
 
 module _ {ℓ ℓ' : Level} (ℂ : Category ℓ ℓ') where
+  private
     open import Cat.Categories.Sets
     open NaturalTransformation (opposite ℂ) (𝓢𝓮𝓽 ℓ')
 
     -- Restrict the functors to Presheafs.
-  RawPresh : RawCategory (ℓ ⊔ lsuc ℓ') (ℓ ⊔ ℓ')
-  RawPresh = record
+    rawPresh : RawCategory (ℓ ⊔ lsuc ℓ') (ℓ ⊔ ℓ')
+    rawPresh = record
       { Object = Presheaf ℂ
       ; Arrow = NaturalTransformation
       ; 𝟙 = λ {F} → identity F
       ; _∘_ = λ {F G H} → NT[_∘_] {F = F} {G = G} {H = H}
       }
+    instance
+      isCategory : IsCategory rawPresh
+      isCategory = Fun.:isCategory: _ _
+
+  Presh : Category (ℓ ⊔ lsuc ℓ') (ℓ ⊔ ℓ')
+  Category.raw        Presh = rawPresh
+  Category.isCategory Presh = isCategory