/-
Copyright (c) 2017 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Tim Baumann, Stephen Morgan, Scott Morrison, Floris van Doorn

! This file was ported from Lean 3 source module category_theory.functor.category
! leanprover-community/mathlib commit 448144f7ae193a8990cb7473c9e9a01990f64ac7
! Please do not edit these lines, except to modify the commit id
! if you have ported upstream changes.
-/
import Mathbin.CategoryTheory.NaturalTransformation
import Mathbin.CategoryTheory.Isomorphism

/-!
# The category of functors and natural transformations between two fixed categories.

> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.
> Any changes to this file require a corresponding PR to mathlib4.

We provide the category instance on `C ⥤ D`, with morphisms the natural transformations.

## Universes

If `C` and `D` are both small categories at the same universe level,
this is another small category at that level.
However if `C` and `D` are both large categories at the same universe level,
this is a small category at the next higher level.
-/

namespace CategoryTheory

-- declare the `v`'s first; see `category_theory.category` for an explanation
universe v₁ v₂ v₃ u₁ u₂ u₃

open NatTrans Category CategoryTheory.Functor

variable (C : Type u₁) [Category.{v₁} C] (D : Type u₂) [Category.{v₂} D]

attribute [local simp] vcomp_app

#print CategoryTheory.Functor.category /-
/-- `functor.category C D` gives the category structure on functors and natural transformations
between categories `C` and `D`.

Notice that if `C` and `D` are both small categories at the same universe level,
this is another small category at that level.
However if `C` and `D` are both large categories at the same universe level,
this is a small category at the next higher level.
-/
instance Functor.category : Category.{max u₁ v₂} (C ⥤ D)
    where
  Hom F G := NatTrans F G
  id F := NatTrans.id F
  comp _ _ _ α β := vcomp α β
#align category_theory.functor.category CategoryTheory.Functor.category
-/

variable {C D} {E : Type u₃} [Category.{v₃} E]

variable {F G H I : C ⥤ D}

namespace NatTrans

#print CategoryTheory.NatTrans.vcomp_eq_comp /-
@[simp]
theorem vcomp_eq_comp (α : F ⟶ G) (β : G ⟶ H) : vcomp α β = α ≫ β :=
  rfl
#align category_theory.nat_trans.vcomp_eq_comp CategoryTheory.NatTrans.vcomp_eq_comp
-/

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theorem vcomp_app' (α : F ⟶ G) (β : G ⟶ H) (X : C) : (α ≫ β).app X = α.app X ≫ β.app X :=
  rfl
#align category_theory.nat_trans.vcomp_app' CategoryTheory.NatTrans.vcomp_app'

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theorem congr_app {α β : F ⟶ G} (h : α = β) (X : C) : α.app X = β.app X := by rw [h]
#align category_theory.nat_trans.congr_app CategoryTheory.NatTrans.congr_app

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@[simp]
theorem id_app (F : C ⥤ D) (X : C) : (𝟙 F : F ⟶ F).app X = 𝟙 (F.obj X) :=
  rfl
#align category_theory.nat_trans.id_app CategoryTheory.NatTrans.id_app

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Case conversion may be inaccurate. Consider using '#align category_theory.nat_trans.comp_app CategoryTheory.NatTrans.comp_appₓ'. -/
@[simp]
theorem comp_app {F G H : C ⥤ D} (α : F ⟶ G) (β : G ⟶ H) (X : C) :
    (α ≫ β).app X = α.app X ≫ β.app X :=
  rfl
#align category_theory.nat_trans.comp_app CategoryTheory.NatTrans.comp_app

/- warning: category_theory.nat_trans.app_naturality -> CategoryTheory.NatTrans.app_naturality is a dubious translation:
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Case conversion may be inaccurate. Consider using '#align category_theory.nat_trans.app_naturality CategoryTheory.NatTrans.app_naturalityₓ'. -/
theorem app_naturality {F G : C ⥤ D ⥤ E} (T : F ⟶ G) (X : C) {Y Z : D} (f : Y ⟶ Z) :
    (F.obj X).map f ≫ (T.app X).app Z = (T.app X).app Y ≫ (G.obj X).map f :=
  (T.app X).naturality f
#align category_theory.nat_trans.app_naturality CategoryTheory.NatTrans.app_naturality

/- warning: category_theory.nat_trans.naturality_app -> CategoryTheory.NatTrans.naturality_app is a dubious translation:
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Case conversion may be inaccurate. Consider using '#align category_theory.nat_trans.naturality_app CategoryTheory.NatTrans.naturality_appₓ'. -/
theorem naturality_app {F G : C ⥤ D ⥤ E} (T : F ⟶ G) (Z : D) {X Y : C} (f : X ⟶ Y) :
    (F.map f).app Z ≫ (T.app Y).app Z = (T.app X).app Z ≫ (G.map f).app Z :=
  congr_fun (congr_arg app (T.naturality f)) Z
#align category_theory.nat_trans.naturality_app CategoryTheory.NatTrans.naturality_app

/- warning: category_theory.nat_trans.mono_of_mono_app -> CategoryTheory.NatTrans.mono_of_mono_app is a dubious translation:
lean 3 declaration is
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but is expected to have type
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Case conversion may be inaccurate. Consider using '#align category_theory.nat_trans.mono_of_mono_app CategoryTheory.NatTrans.mono_of_mono_appₓ'. -/
/-- A natural transformation is a monomorphism if each component is. -/
theorem mono_of_mono_app (α : F ⟶ G) [∀ X : C, Mono (α.app X)] : Mono α :=
  ⟨fun H g h eq => by
    ext X
    rw [← cancel_mono (α.app X), ← comp_app, Eq, comp_app]⟩
#align category_theory.nat_trans.mono_of_mono_app CategoryTheory.NatTrans.mono_of_mono_app

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Case conversion may be inaccurate. Consider using '#align category_theory.nat_trans.epi_of_epi_app CategoryTheory.NatTrans.epi_of_epi_appₓ'. -/
/-- A natural transformation is an epimorphism if each component is. -/
theorem epi_of_epi_app (α : F ⟶ G) [∀ X : C, Epi (α.app X)] : Epi α :=
  ⟨fun H g h eq => by
    ext X
    rw [← cancel_epi (α.app X), ← comp_app, Eq, comp_app]⟩
#align category_theory.nat_trans.epi_of_epi_app CategoryTheory.NatTrans.epi_of_epi_app

#print CategoryTheory.NatTrans.hcomp /-
/-- `hcomp α β` is the horizontal composition of natural transformations. -/
@[simps]
def hcomp {H I : D ⥤ E} (α : F ⟶ G) (β : H ⟶ I) : F ⋙ H ⟶ G ⋙ I
    where
  app := fun X : C => β.app (F.obj X) ≫ I.map (α.app X)
  naturality' X Y f := by
    rw [functor.comp_map, functor.comp_map, ← assoc, naturality, assoc, ← map_comp I, naturality,
      map_comp, assoc]
#align category_theory.nat_trans.hcomp CategoryTheory.NatTrans.hcomp
-/

-- mathport name: «expr ◫ »
infixl:80 " ◫ " => hcomp

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Case conversion may be inaccurate. Consider using '#align category_theory.nat_trans.hcomp_id_app CategoryTheory.NatTrans.hcomp_id_appₓ'. -/
@[simp]
theorem hcomp_id_app {H : D ⥤ E} (α : F ⟶ G) (X : C) : (α ◫ 𝟙 H).app X = H.map (α.app X) :=
  by
  dsimp
  simp
#align category_theory.nat_trans.hcomp_id_app CategoryTheory.NatTrans.hcomp_id_app

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Case conversion may be inaccurate. Consider using '#align category_theory.nat_trans.id_hcomp_app CategoryTheory.NatTrans.id_hcomp_appₓ'. -/
-- See note [dsimp, simp].
theorem id_hcomp_app {H : E ⥤ C} (α : F ⟶ G) (X : E) : (𝟙 H ◫ α).app X = α.app _ := by simp
#align category_theory.nat_trans.id_hcomp_app CategoryTheory.NatTrans.id_hcomp_app

#print CategoryTheory.NatTrans.exchange /-
-- Note that we don't yet prove a `hcomp_assoc` lemma here: even stating it is painful, because we
-- need to use associativity of functor composition. (It's true without the explicit associator,
-- because functor composition is definitionally associative,
-- but relying on the definitional equality causes bad problems with elaboration later.)
theorem exchange {I J K : D ⥤ E} (α : F ⟶ G) (β : G ⟶ H) (γ : I ⟶ J) (δ : J ⟶ K) :
    (α ≫ β) ◫ (γ ≫ δ) = (α ◫ γ) ≫ β ◫ δ := by ext <;> simp
#align category_theory.nat_trans.exchange CategoryTheory.NatTrans.exchange
-/

end NatTrans

open NatTrans

namespace Functor

#print CategoryTheory.Functor.flip /-
/-- Flip the arguments of a bifunctor. See also `currying.lean`. -/
@[simps]
protected def flip (F : C ⥤ D ⥤ E) : D ⥤ C ⥤ E
    where
  obj k :=
    { obj := fun j => (F.obj j).obj k
      map := fun j j' f => (F.map f).app k
      map_id' := fun X => by rw [CategoryTheory.Functor.map_id]; rfl
      map_comp' := fun X Y Z f g => by rw [map_comp, ← comp_app] }
  map c c' f := { app := fun j => (F.obj j).map f }
#align category_theory.functor.flip CategoryTheory.Functor.flip
-/

end Functor

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Case conversion may be inaccurate. Consider using '#align category_theory.map_hom_inv_app CategoryTheory.map_hom_inv_appₓ'. -/
@[simp, reassoc.1]
theorem map_hom_inv_app (F : C ⥤ D ⥤ E) {X Y : C} (e : X ≅ Y) (Z : D) :
    (F.map e.Hom).app Z ≫ (F.map e.inv).app Z = 𝟙 _ := by
  simp [← nat_trans.comp_app, ← functor.map_comp]
#align category_theory.map_hom_inv_app CategoryTheory.map_hom_inv_app

/- warning: category_theory.map_inv_hom_app -> CategoryTheory.map_inv_hom_app is a dubious translation:
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Case conversion may be inaccurate. Consider using '#align category_theory.map_inv_hom_app CategoryTheory.map_inv_hom_appₓ'. -/
@[simp, reassoc.1]
theorem map_inv_hom_app (F : C ⥤ D ⥤ E) {X Y : C} (e : X ≅ Y) (Z : D) :
    (F.map e.inv).app Z ≫ (F.map e.Hom).app Z = 𝟙 _ := by
  simp [← nat_trans.comp_app, ← functor.map_comp]
#align category_theory.map_inv_hom_app CategoryTheory.map_inv_hom_app

end CategoryTheory