Add type-synonyms in functor

src/Cat/Categories/Cat.agda

@@ -16,7 +16,7 @@ open import Cat.Category.Exponential
 open import Cat.Equality
 open Equality.Data.Product
 
-open Functor
+open Functor using (func→ ; func*)
 open Category using (Object ; 𝟙)
 
 -- The category of categories

src/Cat/Category/Functor.agda

@@ -7,7 +7,7 @@ open import Function
 
 open import Cat.Category
 
-open Category hiding (_∘_ ; raw)
+open Category hiding (_∘_ ; raw ; IsIdentity)
 
 module _ {ℓc ℓc' ℓd ℓd'}
     (ℂ : Category ℓc ℓc')
@@ -23,12 +23,18 @@ module _ {ℓc ℓc' ℓd ℓd'}
       func* : Object ℂ → Object 𝔻
       func→ : ∀ {A B} → ℂ [ A , B ] → 𝔻 [ func* A , func* B ]
 
+    IsIdentity : Set _
+    IsIdentity = {A : Object ℂ} → func→ (𝟙 ℂ {A}) ≡ 𝟙 𝔻 {func* A}
+
+    IsDistributive : Set _
+    IsDistributive = {A B C : Object ℂ} {f : ℂ [ A , B ]} {g : ℂ [ B , C ]}
+      → func→ (ℂ [ g ∘ f ]) ≡ 𝔻 [ func→ g ∘ func→ f ]
+
   record IsFunctor (F : RawFunctor) : 𝓤 where
     open RawFunctor F public
     field
-      ident   : {c : Object ℂ} → func→ (𝟙 ℂ {c}) ≡ 𝟙 𝔻 {func* c}
+      ident   : IsIdentity
-      distrib : {A B C : Object ℂ} {f : ℂ [ A , B ]} {g : ℂ [ B , C ]}
+      distrib : IsDistributive
-        → func→ (ℂ [ g ∘ f ]) ≡ 𝔻 [ func→ g ∘ func→ f ]
 
   record Functor : Set (ℓc ⊔ ℓc' ⊔ ℓd ⊔ ℓd') where
     field