Require Import Category.Core Functor.Core.
Require Import FunctorCategory.Core Functor.Composition.Core NaturalTransformation.Composition.Core NaturalTransformation.Composition.Laws.

Set Universe Polymorphism.
Set Implicit Arguments.
Generalizable All Variables.
Set Asymmetric Patterns.

Local Open Scope functor_scope.

Section functorial_composition.
  Context `{Funext}.
  Variables C D E : PreCategory.

  Local Open Scope natural_transformation_scope.

  Definition whiskerL_functor (F : (D → E)%category)
  : ((C → D) → (C → E))%category
    := Build_Functor
         (C → D) (C → E)
         (fun G ⇒ F o G)%functor
         (fun _ _ T ⇒ F oL T)
         (fun _ _ _ _ _ ⇒ composition_of_whisker_l _ _ _)
         (fun _ ⇒ whisker_l_right_identity _ _).

  Definition whiskerR_functor (G : (C → D)%category)
  : ((D → E) → (C → E))%category
    := Build_Functor
         (D → E) (C → E)
         (fun F ⇒ F o G)%functor
         (fun _ _ T ⇒ T oR G)
         (fun _ _ _ _ _ ⇒ composition_of_whisker_r _ _ _)
         (fun _ ⇒ whisker_r_left_identity _ _).
End functorial_composition.