Expose naturalIsProp

src/Cat/Categories/Fun.agda

@@ -38,9 +38,6 @@ module _ {ℓc ℓc' ℓd ℓd' : Level} {ℂ : Category ℓc ℓc'} {𝔻 : Cat
       → (f : ℂ [ A , B ])
       → 𝔻 [ θ B ∘ F.func→ f ] ≡ 𝔻 [ G.func→ f ∘ θ A ]
 
-    -- naturalIsProp : ∀ θ → isProp (Natural θ)
-    -- naturalIsProp θ x y = {!funExt!}
-
     NaturalTransformation : Set (ℓc ⊔ ℓc' ⊔ ℓd')
     NaturalTransformation = Σ Transformation Natural
 
@@ -100,13 +97,11 @@ module _ {ℓc ℓc' ℓd ℓd' : Level} {ℂ : Category ℓc ℓc'} {𝔻 : Cat
     NatComp = _:⊕:_
 
   private
-    module _ {F G : Functor ℂ 𝔻} where
     module 𝔻 = Category 𝔻
 
+  module _ {F G : Functor ℂ 𝔻} where
     transformationIsSet : isSet (Transformation F G)
     transformationIsSet _ _ p q i j C = 𝔻.arrowIsSet _ _ (λ l → p l C)   (λ l → q l C) i j
-      IsSet'   : {ℓ : Level} (A : Set ℓ) → Set ℓ
-      IsSet' A = {x y : A} → (p q : (λ _ → A) [ x ≡ y ]) → p ≡ q
 
     naturalIsProp : (θ : Transformation F G) → isProp (Natural F G θ)
     naturalIsProp θ θNat θNat' = lem
@@ -141,18 +136,17 @@ module _ {ℓc ℓc' ℓd ℓd' : Level} {ℂ : Category ℓc ℓc'} {𝔻 : Cat
       where
         open Category 𝔻
 
-    module _ {A B : Functor ℂ 𝔻} {f : NaturalTransformation A B} where
   private
+    module _ {A B : Functor ℂ 𝔻} {f : NaturalTransformation A B} where
       allNatural = naturalIsProp {F = A} {B}
       f' = proj₁ f
-        module 𝔻Data = Category 𝔻
       eq-r : ∀ C → (𝔻 [ f' C ∘ identityTrans A C ]) ≡ f' C
       eq-r C = begin
         𝔻 [ f' C ∘ identityTrans A C ] ≡⟨⟩
-          𝔻 [ f' C ∘ 𝔻Data.𝟙 ]  ≡⟨ proj₁ (𝔻.ident {A} {B}) ⟩
+        𝔻 [ f' C ∘ 𝔻.𝟙 ]  ≡⟨ proj₁ 𝔻.ident ⟩
         f' C ∎
       eq-l : ∀ C → (𝔻 [ identityTrans B C ∘ f' C ]) ≡ f' C
-        eq-l C = proj₂ (𝔻.ident {A} {B})
+      eq-l C = proj₂ 𝔻.ident
       ident-r : (_:⊕:_ {A} {A} {B} f (identityNat A)) ≡ f
       ident-r = lemSig allNatural _ _ (funExt eq-r)
       ident-l : (_:⊕:_ {A} {B} {B} (identityNat B) f) ≡ f