Move functor-equality to functor module

src/Cat/Categories/Cat.agda

@@ -23,23 +23,6 @@ eqpair eqa eqb i = eqa i , eqb i
 open Functor
 open Category
 
-module _ {ℓ ℓ' : Level} {A B : Category ℓ ℓ'} where
-  lift-eq-functors : {f g : Functor A B}
-    → (eq* : f .func* ≡ g .func*)
-    → (eq→ : PathP (λ i → ∀ {x y} → A .Arrow x y → B .Arrow (eq* i x) (eq* i y))
-    (f .func→) (g .func→))
-    --        → (eq→ : Functor.func→ f ≡ {!!}) -- Functor.func→ g)
-    -- Use PathP
-    -- directly to show heterogeneous equalities by using previous
-    -- equalities (i.e. continuous paths) to create new continuous paths.
-    → (eqI : PathP (λ i → ∀ {c : A .Object} → eq→ i (A .𝟙 {c}) ≡ B .𝟙 {eq* i c})
-    (ident f) (ident g))
-    → (eqD : PathP (λ i → { c c' c'' : A .Object} {a : A .Arrow c c'} {a' : A .Arrow c' c''}
-      → eq→ i (A ._⊕_ a' a) ≡ B ._⊕_ (eq→ i a') (eq→ i a))
-      (distrib f) (distrib g))
-    → f ≡ g
-  lift-eq-functors eq* eq→ eqI eqD i = record { func* = eq* i ; func→ = eq→ i ; ident = eqI i ; distrib = eqD i }
-
 -- The category of categories
 module _ (ℓ ℓ' : Level) where
   private
@@ -59,10 +42,9 @@ module _ (ℓ ℓ' : Level) where
       postulate eqD : PathP (λ i → { c c' c'' : A .Object} {a : A .Arrow c c'} {a' : A .Arrow c' c''}
                         → eq→ i (A ._⊕_ a' a) ≡ D ._⊕_ (eq→ i a') (eq→ i a))
                         (distrib (h ∘f (g ∘f f))) (distrib ((h ∘f g) ∘f f))
-      -- eqD = {!!}
 
       assc : h ∘f (g ∘f f) ≡ (h ∘f g) ∘f f
-      assc = lift-eq-functors eq* eq→ eqI eqD
+      assc = Functor≡ eq* eq→ eqI eqD
 
     module _ {A B : Category ℓ ℓ'} {f : Functor A B} where
       lem : (func* f) ∘ (func* (identity {C = A})) ≡ func* f

src/Cat/Functor.agda

@@ -19,9 +19,29 @@ record Functor {ℓc ℓc' ℓd ℓd'} (C : Category ℓc ℓc') (D : Category
     distrib : { c c' c'' : C .Object} {a : C .Arrow c c'} {a' : C .Arrow c' c''}
       → func→ (C ._⊕_ a' a) ≡ D ._⊕_ (func→ a') (func→ a)
 
-module _ {ℓ ℓ' : Level} {A B C : Category ℓ ℓ'} (F : Functor B C) (G : Functor A B) where
 open Functor
 open Category
+
+module _ {ℓ ℓ' : Level} {ℂ 𝔻 : Category ℓ ℓ'} where
+  private
+    _ℂ⊕_ = ℂ ._⊕_
+  Functor≡ : {F G : Functor ℂ 𝔻}
+    → (eq* : F .func* ≡ G .func*)
+    → (eq→ : PathP (λ i → ∀ {x y} → ℂ .Arrow x y → 𝔻 .Arrow (eq* i x) (eq* i y))
+    (F .func→) (G .func→))
+    --        → (eq→ : Functor.func→ f ≡ {!!}) -- Functor.func→ g)
+    -- Use PathP
+    -- directly to show heterogeneous equalities by using previous
+    -- equalities (i.e. continuous paths) to create new continuous paths.
+    → (eqI : PathP (λ i → ∀ {A : ℂ .Object} → eq→ i (ℂ .𝟙 {A}) ≡ 𝔻 .𝟙 {eq* i A})
+    (ident F) (ident G))
+    → (eqD : PathP (λ i → {A B C : ℂ .Object} {f : ℂ .Arrow A B} {g : ℂ .Arrow B C}
+    → eq→ i (ℂ ._⊕_ g f) ≡ 𝔻 ._⊕_ (eq→ i g) (eq→ i f))
+    (distrib F) (distrib G))
+    → F ≡ G
+  Functor≡ eq* eq→ eqI eqD i = record { func* = eq* i ; func→ = eq→ i ; ident = eqI i ; distrib = eqD i }
+
+module _ {ℓ ℓ' : Level} {A B C : Category ℓ ℓ'} (F : Functor B C) (G : Functor A B) where
   private
     F* = F .func*
     F→ = F .func→