(** * Functoriality of functor composition *)
Require Import Category.Core Functor.Core NaturalTransformation.Core.
[Loading ML file number_string_notation_plugin.cmxs (using legacy method) ... done]
Require Import Functor.Composition.Core NaturalTransformation.Composition.Core.
Require Import Category.Prod FunctorCategory.Core NaturalTransformation.Composition.Functorial NaturalTransformation.Composition.Laws ExponentialLaws.Law4.Functors.
Require Import NaturalTransformation.Paths.
Require ProductLaws.

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

(** Construction of the functor [_∘_ : (C → D) × (D → E) → (C → E)] and its curried variant *)
Section functorial_composition.
  Context `{Funext}.
  Variables C D E : PreCategory.

  Local Open Scope natural_transformation_scope.

  Definition compose_functor_morphism_of
             s d (m : morphism (C -> D) s d)
  : morphism ((D -> E) -> (C -> E))
             (whiskerR_functor _ s)
             (whiskerR_functor _ d)
    := Build_NaturalTransformation
         (whiskerR_functor E s)
         (whiskerR_functor E d)
         (fun x => x oL m)
         (fun _ _ _ => exchange_whisker _ _).

  Definition compose_functor : object ((C -> D) -> ((D -> E) -> (C -> E))).
H: Funext
C, D, E: PreCategory
((C -> D) -> (D -> E) -> C -> E)%category
  Proof.
H: Funext
C, D, E: PreCategory
((C -> D) -> (D -> E) -> C -> E)%category
    refine (Build_Functor
              (C -> D) ((D -> E) -> (C -> E))
              (@whiskerR_functor _ _ _ _)
              compose_functor_morphism_of
              _
              _);
    abstract (
        path_natural_transformation;
        rewrite ?composition_of, ?identity_of;
        reflexivity
      ).
  Defined.

  Definition compose_functor_uncurried
  : object ((C -> D) * (D -> E) -> (C -> E))
    := ExponentialLaws.Law4.Functors.functor _ _ _ compose_functor.

  Definition compose_functor' : object ((D -> E) -> ((C -> D) -> (C -> E)))
    := ExponentialLaws.Law4.Functors.inverse
         _ _ _ (compose_functor_uncurried o ProductLaws.Swap.functor _ _)%functor.
End functorial_composition.