Require Import Category.Core Functor.Core Category.Prod Functor.Prod.Core NaturalTransformation.Core.
Require Import Types.Prod.

Set Implicit Arguments.
Generalizable All Variables.

Section prod.
  Context {A : PreCategory}.
  Context {B : PreCategory}.
  Context {C : PreCategory}.
  Variables F F' : Functor A B.
  Variables G G' : Functor A C.
  Variable T : NaturalTransformation F F'.
  Variable U : NaturalTransformation G G'.

  Definition prod
  : NaturalTransformation (F × G) (F' × G')
    := Build_NaturalTransformation
         (F × G) (F' × G')
         (fun x : A ⇒ (T x, U x))
         (fun _ _ _ ⇒ path_prod' (commutes T _ _ _) (commutes U _ _ _)).
End prod.

Local Infix "×" := prod : natural_transformation_scope.

Section induced.
  Variables C D E : PreCategory.

  Variable F : Functor (C × D) E.

  Local Ltac t :=
    simpl; intros;
    rewrite <- !composition_of;
    simpl;
    rewrite ?left_identity, ?right_identity;
    reflexivity.

  Definition induced_fst s d (m : morphism C s d)
  : NaturalTransformation (Functor.Prod.Core.induced_snd F s)
                          (Functor.Prod.Core.induced_snd F d).
  Proof.
    let F0 := match goal with |- NaturalTransformation ?F0 ?G0 ⇒ constr:(F0) end in
    let G0 := match goal with |- NaturalTransformation ?F0 ?G0 ⇒ constr:(G0) end in
    refine (Build_NaturalTransformation
              F0 G0
              (fun d ⇒ @morphism_of _ _ F (_, _) (_, _) (m, @identity D d))
              _).
    abstract t.
  Defined.

  Definition induced_snd s d (m : morphism D s d)
  : NaturalTransformation (Functor.Prod.Core.induced_fst F s)
                          (Functor.Prod.Core.induced_fst F d).
  Proof.
    let F0 := match goal with |- NaturalTransformation ?F0 ?G0 ⇒ constr:(F0) end in
    let G0 := match goal with |- NaturalTransformation ?F0 ?G0 ⇒ constr:(G0) end in
    refine (Build_NaturalTransformation
              F0 G0
              (fun c ⇒ @morphism_of _ _ F (_, _) (_, _) (@identity C c, m))
              _).
    abstract t.
  Defined.
End induced.

Module Export NaturalTransformationProdNotations.
  Infix "×" := prod : natural_transformation_scope.
End NaturalTransformationProdNotations.