Rename natural transformation composition

src/Cat/Categories/Fun.agda

@@ -55,11 +55,11 @@ module Fun {ℓc ℓc' ℓd ℓd' : Level} (ℂ : Category ℓc ℓc') (𝔻 : C
       ηNat = proj₂ η'
       ζNat = proj₂ ζ'
       L : NaturalTransformation A D
-      L = (_:⊕:_ {A} {C} {D} ζ' (_:⊕:_ {A} {B} {C} η' θ'))
+      L = (NT[_∘_] {A} {C} {D} ζ' (NT[_∘_] {A} {B} {C} η' θ'))
       R : NaturalTransformation A D
-      R = (_:⊕:_ {A} {B} {D} (_:⊕:_ {B} {C} {D} ζ' η') θ')
+      R = (NT[_∘_] {A} {B} {D} (NT[_∘_] {B} {C} {D} ζ' η') θ')
-    _g⊕f_ = _:⊕:_ {A} {B} {C}
+    _g⊕f_ = NT[_∘_] {A} {B} {C}
-    _h⊕g_ = _:⊕:_ {B} {C} {D}
+    _h⊕g_ = NT[_∘_] {B} {C} {D}
     :isAssociative: : L ≡ R
     :isAssociative: = lemSig (naturalIsProp {F = A} {D})
       L R (funExt (λ x → isAssociative))
@@ -77,13 +77,13 @@ module Fun {ℓc ℓc' ℓd ℓd' : Level} (ℂ : Category ℓc ℓc') (𝔻 : C
         f' C ∎
       eq-l : ∀ C → (𝔻 [ identityTrans B C ∘ f' C ]) ≡ f' C
       eq-l C = proj₂ 𝔻.isIdentity
-      ident-r : (_:⊕:_ {A} {A} {B} f (NT.identity A)) ≡ f
+      ident-r : (NT[_∘_] {A} {A} {B} f (NT.identity A)) ≡ f
       ident-r = lemSig allNatural _ _ (funExt eq-r)
-      ident-l : (_:⊕:_ {A} {B} {B} (NT.identity B) f) ≡ f
+      ident-l : (NT[_∘_] {A} {B} {B} (NT.identity B) f) ≡ f
       ident-l = lemSig allNatural _ _ (funExt eq-l)
       isIdentity
-        : (_:⊕:_ {A} {A} {B} f (NT.identity A)) ≡ f
+        : (NT[_∘_] {A} {A} {B} f (NT.identity A)) ≡ f
-        × (_:⊕:_ {A} {B} {B} (NT.identity B) f) ≡ f
+        × (NT[_∘_] {A} {B} {B} (NT.identity B) f) ≡ f
       isIdentity = ident-r , ident-l
   -- Functor categories. Objects are functors, arrows are natural transformations.
   RawFun : RawCategory (ℓc ⊔ ℓc' ⊔ ℓd ⊔ ℓd') (ℓc ⊔ ℓc' ⊔ ℓd')
@@ -91,7 +91,7 @@ module Fun {ℓc ℓc' ℓd ℓd' : Level} (ℂ : Category ℓc ℓc') (𝔻 : C
     { Object = Functor ℂ 𝔻
     ; Arrow = NaturalTransformation
     ; 𝟙 = λ {F} → NT.identity F
-    ; _∘_ = λ {F G H} → _:⊕:_ {F} {G} {H}
+    ; _∘_ = λ {F G H} → NT[_∘_] {F} {G} {H}
     }
 
   instance
@@ -116,5 +116,5 @@ module _ {ℓ ℓ' : Level} (ℂ : Category ℓ ℓ') where
     { Object = Presheaf ℂ
     ; Arrow = NaturalTransformation
     ; 𝟙 = λ {F} → identity F
-    ; _∘_ = λ {F G H} → NatComp {F = F} {G = G} {H = H}
+    ; _∘_ = λ {F G H} → NT[_∘_] {F = F} {G = G} {H = H}
     }

src/Cat/Category/NaturalTransformation.agda

@@ -83,21 +83,19 @@ module NaturalTransformation {ℓc ℓc' ℓd ℓd' : Level}
       module F = Functor F
       module G = Functor G
       module H = Functor H
-      _∘nt_ : Transformation G H → Transformation F G → Transformation F H
+    T[_∘_] : Transformation G H → Transformation F G → Transformation F H
-      (θ ∘nt η) C = 𝔻 [ θ C ∘ η C ]
+    T[ θ ∘ η ] C = 𝔻 [ θ C ∘ η C ]
 
-    NatComp _:⊕:_ : NaturalTransformation G H → NaturalTransformation F G → NaturalTransformation F H
+    NT[_∘_] : NaturalTransformation G H → NaturalTransformation F G → NaturalTransformation F H
-    proj₁ ((θ , _) :⊕: (η , _)) = θ ∘nt η
+    proj₁ NT[ (θ , _) ∘ (η , _) ] = T[ θ ∘ η ]
-    proj₂ ((θ , θNat) :⊕: (η , ηNat)) {A} {B} f = begin
+    proj₂ NT[ (θ , θNat) ∘ (η , ηNat) ] {A} {B} f = begin
-      𝔻 [ (θ ∘nt η) B ∘ F.func→ f ]     ≡⟨⟩
+      𝔻 [ T[ θ ∘ η ] B ∘ F.func→ f ]     ≡⟨⟩
       𝔻 [ 𝔻 [ θ B ∘ η B ] ∘ F.func→ f ] ≡⟨ sym isAssociative ⟩
       𝔻 [ θ B ∘ 𝔻 [ η B ∘ F.func→ f ] ] ≡⟨ cong (λ φ → 𝔻 [ θ B ∘ φ ]) (ηNat f) ⟩
       𝔻 [ θ B ∘ 𝔻 [ G.func→ f ∘ η A ] ] ≡⟨ isAssociative ⟩
       𝔻 [ 𝔻 [ θ B ∘ G.func→ f ] ∘ η A ] ≡⟨ cong (λ φ → 𝔻 [ φ ∘ η A ]) (θNat f) ⟩
       𝔻 [ 𝔻 [ H.func→ f ∘ θ A ] ∘ η A ] ≡⟨ sym isAssociative ⟩
       𝔻 [ H.func→ f ∘ 𝔻 [ θ A ∘ η A ] ] ≡⟨⟩
-      𝔻 [ H.func→ f ∘ (θ ∘nt η) A ]     ∎
+      𝔻 [ H.func→ f ∘ T[ θ ∘ η ] A ]     ∎
       where
         open Category 𝔻
-
-    NatComp = _:⊕:_