/-
Copyright (c) 2020 Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta, Thomas Read, Andrew Yang

! This file was ported from Lean 3 source module category_theory.adjunction.opposites
! leanprover-community/mathlib commit 0148d455199ed64bf8eb2f493a1e7eb9211ce170
! Please do not edit these lines, except to modify the commit id
! if you have ported upstream changes.
-/
import Mathbin.CategoryTheory.Adjunction.Basic
import Mathbin.CategoryTheory.Yoneda
import Mathbin.CategoryTheory.Opposites

/-!
# Opposite adjunctions

This file contains constructions to relate adjunctions of functors to adjunctions of their
opposites.
These constructions are used to show uniqueness of adjoints (up to natural isomorphism).

## Tags
adjunction, opposite, uniqueness
-/

open CategoryTheory

universe v₁ v₂ u₁ u₂

-- morphism levels before object levels. See note [category_theory universes].
variable {C : Type u₁} [Category.{v₁} C] {D : Type u₂} [Category.{v₂} D]

namespace CategoryTheory.Adjunction

#print CategoryTheory.Adjunction.adjointOfOpAdjointOp /-
/-- If `G.op` is adjoint to `F.op` then `F` is adjoint to `G`. -/
@[simps unit_app counit_app]
def adjointOfOpAdjointOp (F : C ⥤ D) (G : D ⥤ C) (h : G.op ⊣ F.op) : F ⊣ G :=
  Adjunction.mkOfHomEquiv
    {
      homEquiv := fun X Y =>
        ((h.homEquiv (Opposite.op Y) (Opposite.op X)).trans (opEquiv _ _)).symm.trans
          (opEquiv _ _) }
#align category_theory.adjunction.adjoint_of_op_adjoint_op CategoryTheory.Adjunction.adjointOfOpAdjointOp
-/

#print CategoryTheory.Adjunction.adjointUnopOfAdjointOp /-
/-- If `G` is adjoint to `F.op` then `F` is adjoint to `G.unop`. -/
def adjointUnopOfAdjointOp (F : C ⥤ D) (G : Dᵒᵖ ⥤ Cᵒᵖ) (h : G ⊣ F.op) : F ⊣ G.unop :=
  adjointOfOpAdjointOp F G.unop (h.ofNatIsoLeft G.opUnopIso.symm)
#align category_theory.adjunction.adjoint_unop_of_adjoint_op CategoryTheory.Adjunction.adjointUnopOfAdjointOp
-/

#print CategoryTheory.Adjunction.unopAdjointOfOpAdjoint /-
/-- If `G.op` is adjoint to `F` then `F.unop` is adjoint to `G`. -/
def unopAdjointOfOpAdjoint (F : Cᵒᵖ ⥤ Dᵒᵖ) (G : D ⥤ C) (h : G.op ⊣ F) : F.unop ⊣ G :=
  adjointOfOpAdjointOp _ _ (h.ofNatIsoRight F.opUnopIso.symm)
#align category_theory.adjunction.unop_adjoint_of_op_adjoint CategoryTheory.Adjunction.unopAdjointOfOpAdjoint
-/

#print CategoryTheory.Adjunction.unopAdjointUnopOfAdjoint /-
/-- If `G` is adjoint to `F` then `F.unop` is adjoint to `G.unop`. -/
def unopAdjointUnopOfAdjoint (F : Cᵒᵖ ⥤ Dᵒᵖ) (G : Dᵒᵖ ⥤ Cᵒᵖ) (h : G ⊣ F) : F.unop ⊣ G.unop :=
  adjointUnopOfAdjointOp F.unop G (h.ofNatIsoRight F.opUnopIso.symm)
#align category_theory.adjunction.unop_adjoint_unop_of_adjoint CategoryTheory.Adjunction.unopAdjointUnopOfAdjoint
-/

#print CategoryTheory.Adjunction.opAdjointOpOfAdjoint /-
/-- If `G` is adjoint to `F` then `F.op` is adjoint to `G.op`. -/
@[simps unit_app counit_app]
def opAdjointOpOfAdjoint (F : C ⥤ D) (G : D ⥤ C) (h : G ⊣ F) : F.op ⊣ G.op :=
  Adjunction.mkOfHomEquiv
    {
      homEquiv := fun X Y =>
        (opEquiv _ Y).trans ((h.homEquiv _ _).symm.trans (opEquiv X (Opposite.op _)).symm) }
#align category_theory.adjunction.op_adjoint_op_of_adjoint CategoryTheory.Adjunction.opAdjointOpOfAdjoint
-/

#print CategoryTheory.Adjunction.adjointOpOfAdjointUnop /-
/-- If `G` is adjoint to `F.unop` then `F` is adjoint to `G.op`. -/
def adjointOpOfAdjointUnop (F : Cᵒᵖ ⥤ Dᵒᵖ) (G : D ⥤ C) (h : G ⊣ F.unop) : F ⊣ G.op :=
  (opAdjointOpOfAdjoint F.unop _ h).ofNatIsoLeft F.opUnopIso
#align category_theory.adjunction.adjoint_op_of_adjoint_unop CategoryTheory.Adjunction.adjointOpOfAdjointUnop
-/

#print CategoryTheory.Adjunction.opAdjointOfUnopAdjoint /-
/-- If `G.unop` is adjoint to `F` then `F.op` is adjoint to `G`. -/
def opAdjointOfUnopAdjoint (F : C ⥤ D) (G : Dᵒᵖ ⥤ Cᵒᵖ) (h : G.unop ⊣ F) : F.op ⊣ G :=
  (opAdjointOpOfAdjoint _ G.unop h).ofNatIsoRight G.opUnopIso
#align category_theory.adjunction.op_adjoint_of_unop_adjoint CategoryTheory.Adjunction.opAdjointOfUnopAdjoint
-/

#print CategoryTheory.Adjunction.adjointOfUnopAdjointUnop /-
/-- If `G.unop` is adjoint to `F.unop` then `F` is adjoint to `G`. -/
def adjointOfUnopAdjointUnop (F : Cᵒᵖ ⥤ Dᵒᵖ) (G : Dᵒᵖ ⥤ Cᵒᵖ) (h : G.unop ⊣ F.unop) : F ⊣ G :=
  (adjointOpOfAdjointUnop _ _ h).ofNatIsoRight G.opUnopIso
#align category_theory.adjunction.adjoint_of_unop_adjoint_unop CategoryTheory.Adjunction.adjointOfUnopAdjointUnop
-/

#print CategoryTheory.Adjunction.leftAdjointsCoyonedaEquiv /-
/-- If `F` and `F'` are both adjoint to `G`, there is a natural isomorphism
`F.op ⋙ coyoneda ≅ F'.op ⋙ coyoneda`.
We use this in combination with `fully_faithful_cancel_right` to show left adjoints are unique.
-/
def leftAdjointsCoyonedaEquiv {F F' : C ⥤ D} {G : D ⥤ C} (adj1 : F ⊣ G) (adj2 : F' ⊣ G) :
    F.op ⋙ coyoneda ≅ F'.op ⋙ coyoneda :=
  NatIso.ofComponents
    (fun X =>
      NatIso.ofComponents
        (fun Y => ((adj1.homEquiv X.unop Y).trans (adj2.homEquiv X.unop Y).symm).toIso) (by tidy))
    (by tidy)
#align category_theory.adjunction.left_adjoints_coyoneda_equiv CategoryTheory.Adjunction.leftAdjointsCoyonedaEquiv
-/

#print CategoryTheory.Adjunction.leftAdjointUniq /-
/-- If `F` and `F'` are both left adjoint to `G`, then they are naturally isomorphic. -/
def leftAdjointUniq {F F' : C ⥤ D} {G : D ⥤ C} (adj1 : F ⊣ G) (adj2 : F' ⊣ G) : F ≅ F' :=
  NatIso.removeOp (fullyFaithfulCancelRight _ (leftAdjointsCoyonedaEquiv adj2 adj1))
#align category_theory.adjunction.left_adjoint_uniq CategoryTheory.Adjunction.leftAdjointUniq
-/

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Case conversion may be inaccurate. Consider using '#align category_theory.adjunction.hom_equiv_left_adjoint_uniq_hom_app CategoryTheory.Adjunction.homEquiv_leftAdjointUniq_hom_appₓ'. -/
@[simp]
theorem homEquiv_leftAdjointUniq_hom_app {F F' : C ⥤ D} {G : D ⥤ C} (adj1 : F ⊣ G) (adj2 : F' ⊣ G)
    (x : C) : adj1.homEquiv _ _ ((leftAdjointUniq adj1 adj2).Hom.app x) = adj2.Unit.app x :=
  by
  apply (adj1.hom_equiv _ _).symm.Injective
  apply Quiver.Hom.op_inj
  apply coyoneda.map_injective
  swap; infer_instance
  ext (f y)
  simpa [left_adjoint_uniq, left_adjoints_coyoneda_equiv]
#align category_theory.adjunction.hom_equiv_left_adjoint_uniq_hom_app CategoryTheory.Adjunction.homEquiv_leftAdjointUniq_hom_app

#print CategoryTheory.Adjunction.unit_leftAdjointUniq_hom /-
@[simp, reassoc.1]
theorem unit_leftAdjointUniq_hom {F F' : C ⥤ D} {G : D ⥤ C} (adj1 : F ⊣ G) (adj2 : F' ⊣ G) :
    adj1.Unit ≫ whiskerRight (leftAdjointUniq adj1 adj2).Hom G = adj2.Unit :=
  by
  ext x
  rw [nat_trans.comp_app, ← hom_equiv_left_adjoint_uniq_hom_app adj1 adj2]
  simp [-hom_equiv_left_adjoint_uniq_hom_app, ← G.map_comp]
#align category_theory.adjunction.unit_left_adjoint_uniq_hom CategoryTheory.Adjunction.unit_leftAdjointUniq_hom
-/

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Case conversion may be inaccurate. Consider using '#align category_theory.adjunction.unit_left_adjoint_uniq_hom_app CategoryTheory.Adjunction.unit_leftAdjointUniq_hom_appₓ'. -/
@[simp, reassoc.1]
theorem unit_leftAdjointUniq_hom_app {F F' : C ⥤ D} {G : D ⥤ C} (adj1 : F ⊣ G) (adj2 : F' ⊣ G)
    (x : C) : adj1.Unit.app x ≫ G.map ((leftAdjointUniq adj1 adj2).Hom.app x) = adj2.Unit.app x :=
  by
  rw [← unit_left_adjoint_uniq_hom adj1 adj2]
  rfl
#align category_theory.adjunction.unit_left_adjoint_uniq_hom_app CategoryTheory.Adjunction.unit_leftAdjointUniq_hom_app

#print CategoryTheory.Adjunction.leftAdjointUniq_hom_counit /-
@[simp, reassoc.1]
theorem leftAdjointUniq_hom_counit {F F' : C ⥤ D} {G : D ⥤ C} (adj1 : F ⊣ G) (adj2 : F' ⊣ G) :
    whiskerLeft G (leftAdjointUniq adj1 adj2).Hom ≫ adj2.counit = adj1.counit :=
  by
  ext x
  apply Quiver.Hom.op_inj
  apply coyoneda.map_injective
  swap
  infer_instance
  ext (y f)
  have :
    F.map (adj2.unit.app (G.obj x)) ≫ adj1.counit.app (F'.obj (G.obj x)) ≫ adj2.counit.app x ≫ f =
      adj1.counit.app x ≫ f :=
    by
    erw [← adj1.counit.naturality, ← F.map_comp_assoc]
    simpa
  simpa [left_adjoint_uniq, left_adjoints_coyoneda_equiv] using this
#align category_theory.adjunction.left_adjoint_uniq_hom_counit CategoryTheory.Adjunction.leftAdjointUniq_hom_counit
-/

/- warning: category_theory.adjunction.left_adjoint_uniq_hom_app_counit -> CategoryTheory.Adjunction.leftAdjointUniq_hom_app_counit is a dubious translation:
lean 3 declaration is
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Case conversion may be inaccurate. Consider using '#align category_theory.adjunction.left_adjoint_uniq_hom_app_counit CategoryTheory.Adjunction.leftAdjointUniq_hom_app_counitₓ'. -/
@[simp, reassoc.1]
theorem leftAdjointUniq_hom_app_counit {F F' : C ⥤ D} {G : D ⥤ C} (adj1 : F ⊣ G) (adj2 : F' ⊣ G)
    (x : D) :
    (leftAdjointUniq adj1 adj2).Hom.app (G.obj x) ≫ adj2.counit.app x = adj1.counit.app x :=
  by
  rw [← left_adjoint_uniq_hom_counit adj1 adj2]
  rfl
#align category_theory.adjunction.left_adjoint_uniq_hom_app_counit CategoryTheory.Adjunction.leftAdjointUniq_hom_app_counit

/- warning: category_theory.adjunction.left_adjoint_uniq_inv_app -> CategoryTheory.Adjunction.leftAdjointUniq_inv_app is a dubious translation:
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Case conversion may be inaccurate. Consider using '#align category_theory.adjunction.left_adjoint_uniq_inv_app CategoryTheory.Adjunction.leftAdjointUniq_inv_appₓ'. -/
@[simp]
theorem leftAdjointUniq_inv_app {F F' : C ⥤ D} {G : D ⥤ C} (adj1 : F ⊣ G) (adj2 : F' ⊣ G) (x : C) :
    (leftAdjointUniq adj1 adj2).inv.app x = (leftAdjointUniq adj2 adj1).Hom.app x :=
  rfl
#align category_theory.adjunction.left_adjoint_uniq_inv_app CategoryTheory.Adjunction.leftAdjointUniq_inv_app

#print CategoryTheory.Adjunction.leftAdjointUniq_trans /-
@[simp, reassoc.1]
theorem leftAdjointUniq_trans {F F' F'' : C ⥤ D} {G : D ⥤ C} (adj1 : F ⊣ G) (adj2 : F' ⊣ G)
    (adj3 : F'' ⊣ G) :
    (leftAdjointUniq adj1 adj2).Hom ≫ (leftAdjointUniq adj2 adj3).Hom =
      (leftAdjointUniq adj1 adj3).Hom :=
  by
  ext
  apply Quiver.Hom.op_inj
  apply coyoneda.map_injective
  swap; infer_instance
  ext
  simp [left_adjoints_coyoneda_equiv, left_adjoint_uniq]
#align category_theory.adjunction.left_adjoint_uniq_trans CategoryTheory.Adjunction.leftAdjointUniq_trans
-/

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Case conversion may be inaccurate. Consider using '#align category_theory.adjunction.left_adjoint_uniq_trans_app CategoryTheory.Adjunction.leftAdjointUniq_trans_appₓ'. -/
@[simp, reassoc.1]
theorem leftAdjointUniq_trans_app {F F' F'' : C ⥤ D} {G : D ⥤ C} (adj1 : F ⊣ G) (adj2 : F' ⊣ G)
    (adj3 : F'' ⊣ G) (x : C) :
    (leftAdjointUniq adj1 adj2).Hom.app x ≫ (leftAdjointUniq adj2 adj3).Hom.app x =
      (leftAdjointUniq adj1 adj3).Hom.app x :=
  by
  rw [← left_adjoint_uniq_trans adj1 adj2 adj3]
  rfl
#align category_theory.adjunction.left_adjoint_uniq_trans_app CategoryTheory.Adjunction.leftAdjointUniq_trans_app

#print CategoryTheory.Adjunction.leftAdjointUniq_refl /-
@[simp]
theorem leftAdjointUniq_refl {F : C ⥤ D} {G : D ⥤ C} (adj1 : F ⊣ G) :
    (leftAdjointUniq adj1 adj1).Hom = 𝟙 _ := by
  ext
  apply Quiver.Hom.op_inj
  apply coyoneda.map_injective
  swap; infer_instance
  ext
  simp [left_adjoints_coyoneda_equiv, left_adjoint_uniq]
#align category_theory.adjunction.left_adjoint_uniq_refl CategoryTheory.Adjunction.leftAdjointUniq_refl
-/

#print CategoryTheory.Adjunction.rightAdjointUniq /-
/-- If `G` and `G'` are both right adjoint to `F`, then they are naturally isomorphic. -/
def rightAdjointUniq {F : C ⥤ D} {G G' : D ⥤ C} (adj1 : F ⊣ G) (adj2 : F ⊣ G') : G ≅ G' :=
  NatIso.removeOp (leftAdjointUniq (opAdjointOpOfAdjoint _ F adj2) (opAdjointOpOfAdjoint _ _ adj1))
#align category_theory.adjunction.right_adjoint_uniq CategoryTheory.Adjunction.rightAdjointUniq
-/

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Case conversion may be inaccurate. Consider using '#align category_theory.adjunction.hom_equiv_symm_right_adjoint_uniq_hom_app CategoryTheory.Adjunction.homEquiv_symm_rightAdjointUniq_hom_appₓ'. -/
@[simp]
theorem homEquiv_symm_rightAdjointUniq_hom_app {F : C ⥤ D} {G G' : D ⥤ C} (adj1 : F ⊣ G)
    (adj2 : F ⊣ G') (x : D) :
    (adj2.homEquiv _ _).symm ((rightAdjointUniq adj1 adj2).Hom.app x) = adj1.counit.app x :=
  by
  apply Quiver.Hom.op_inj
  convert hom_equiv_left_adjoint_uniq_hom_app (op_adjoint_op_of_adjoint _ F adj2)
      (op_adjoint_op_of_adjoint _ _ adj1) (Opposite.op x)
  simpa
#align category_theory.adjunction.hom_equiv_symm_right_adjoint_uniq_hom_app CategoryTheory.Adjunction.homEquiv_symm_rightAdjointUniq_hom_app

/- warning: category_theory.adjunction.unit_right_adjoint_uniq_hom_app -> CategoryTheory.Adjunction.unit_rightAdjointUniq_hom_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.adjunction.unit_right_adjoint_uniq_hom_app CategoryTheory.Adjunction.unit_rightAdjointUniq_hom_appₓ'. -/
@[simp, reassoc.1]
theorem unit_rightAdjointUniq_hom_app {F : C ⥤ D} {G G' : D ⥤ C} (adj1 : F ⊣ G) (adj2 : F ⊣ G')
    (x : C) : adj1.Unit.app x ≫ (rightAdjointUniq adj1 adj2).Hom.app (F.obj x) = adj2.Unit.app x :=
  by
  apply Quiver.Hom.op_inj
  convert left_adjoint_uniq_hom_app_counit (op_adjoint_op_of_adjoint _ _ adj2)
      (op_adjoint_op_of_adjoint _ _ adj1) (Opposite.op x)
  all_goals simpa
#align category_theory.adjunction.unit_right_adjoint_uniq_hom_app CategoryTheory.Adjunction.unit_rightAdjointUniq_hom_app

#print CategoryTheory.Adjunction.unit_rightAdjointUniq_hom /-
@[simp, reassoc.1]
theorem unit_rightAdjointUniq_hom {F : C ⥤ D} {G G' : D ⥤ C} (adj1 : F ⊣ G) (adj2 : F ⊣ G') :
    adj1.Unit ≫ whiskerLeft F (rightAdjointUniq adj1 adj2).Hom = adj2.Unit :=
  by
  ext x
  simp
#align category_theory.adjunction.unit_right_adjoint_uniq_hom CategoryTheory.Adjunction.unit_rightAdjointUniq_hom
-/

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Case conversion may be inaccurate. Consider using '#align category_theory.adjunction.right_adjoint_uniq_hom_app_counit CategoryTheory.Adjunction.rightAdjointUniq_hom_app_counitₓ'. -/
@[simp, reassoc.1]
theorem rightAdjointUniq_hom_app_counit {F : C ⥤ D} {G G' : D ⥤ C} (adj1 : F ⊣ G) (adj2 : F ⊣ G')
    (x : D) :
    F.map ((rightAdjointUniq adj1 adj2).Hom.app x) ≫ adj2.counit.app x = adj1.counit.app x :=
  by
  apply Quiver.Hom.op_inj
  convert unit_left_adjoint_uniq_hom_app (op_adjoint_op_of_adjoint _ _ adj2)
      (op_adjoint_op_of_adjoint _ _ adj1) (Opposite.op x)
  all_goals simpa
#align category_theory.adjunction.right_adjoint_uniq_hom_app_counit CategoryTheory.Adjunction.rightAdjointUniq_hom_app_counit

#print CategoryTheory.Adjunction.rightAdjointUniq_hom_counit /-
@[simp, reassoc.1]
theorem rightAdjointUniq_hom_counit {F : C ⥤ D} {G G' : D ⥤ C} (adj1 : F ⊣ G) (adj2 : F ⊣ G') :
    whiskerRight (rightAdjointUniq adj1 adj2).Hom F ≫ adj2.counit = adj1.counit :=
  by
  ext
  simp
#align category_theory.adjunction.right_adjoint_uniq_hom_counit CategoryTheory.Adjunction.rightAdjointUniq_hom_counit
-/

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Case conversion may be inaccurate. Consider using '#align category_theory.adjunction.right_adjoint_uniq_inv_app CategoryTheory.Adjunction.rightAdjointUniq_inv_appₓ'. -/
@[simp]
theorem rightAdjointUniq_inv_app {F : C ⥤ D} {G G' : D ⥤ C} (adj1 : F ⊣ G) (adj2 : F ⊣ G') (x : D) :
    (rightAdjointUniq adj1 adj2).inv.app x = (rightAdjointUniq adj2 adj1).Hom.app x :=
  rfl
#align category_theory.adjunction.right_adjoint_uniq_inv_app CategoryTheory.Adjunction.rightAdjointUniq_inv_app

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but is expected to have type
  forall {C : Type.{u3}} [_inst_1 : CategoryTheory.Category.{u1, u3} C] {D : Type.{u4}} [_inst_2 : CategoryTheory.Category.{u2, u4} D] {F : CategoryTheory.Functor.{u1, u2, u3, u4} C _inst_1 D _inst_2} {G : CategoryTheory.Functor.{u2, u1, u4, u3} D _inst_2 C _inst_1} {G' : CategoryTheory.Functor.{u2, u1, u4, u3} D _inst_2 C _inst_1} {G'' : CategoryTheory.Functor.{u2, u1, u4, u3} D _inst_2 C _inst_1} (adj1 : CategoryTheory.Adjunction.{u1, u2, u3, u4} C _inst_1 D _inst_2 F G) (adj2 : CategoryTheory.Adjunction.{u1, u2, u3, u4} C _inst_1 D _inst_2 F G') (adj3 : CategoryTheory.Adjunction.{u1, u2, u3, u4} C _inst_1 D _inst_2 F G'') (x : D), Eq.{succ u1} (Quiver.Hom.{succ u1, u3} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) (Prefunctor.obj.{succ u2, succ u1, u4, u3} D (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} D (CategoryTheory.Category.toCategoryStruct.{u2, u4} D _inst_2)) C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) (CategoryTheory.Functor.toPrefunctor.{u2, u1, u4, u3} D _inst_2 C _inst_1 G) x) (Prefunctor.obj.{succ u2, succ u1, u4, u3} D (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} D (CategoryTheory.Category.toCategoryStruct.{u2, u4} D _inst_2)) C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) (CategoryTheory.Functor.toPrefunctor.{u2, u1, u4, u3} D _inst_2 C _inst_1 G'') x)) (CategoryTheory.CategoryStruct.comp.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1) (Prefunctor.obj.{succ u2, succ u1, u4, u3} D (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} D (CategoryTheory.Category.toCategoryStruct.{u2, u4} D _inst_2)) C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) (CategoryTheory.Functor.toPrefunctor.{u2, u1, u4, u3} D _inst_2 C _inst_1 G) x) (Prefunctor.obj.{succ u2, succ u1, u4, u3} D (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} D (CategoryTheory.Category.toCategoryStruct.{u2, u4} D _inst_2)) C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) (CategoryTheory.Functor.toPrefunctor.{u2, u1, u4, u3} D _inst_2 C _inst_1 G') x) (Prefunctor.obj.{succ u2, succ u1, u4, u3} D (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} D (CategoryTheory.Category.toCategoryStruct.{u2, u4} D _inst_2)) C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) (CategoryTheory.Functor.toPrefunctor.{u2, u1, u4, u3} D _inst_2 C _inst_1 G'') x) (CategoryTheory.NatTrans.app.{u2, u1, u4, u3} D _inst_2 C _inst_1 G G' (CategoryTheory.Iso.hom.{max u4 u1, max (max (max u3 u4) u1) u2} (CategoryTheory.Functor.{u2, u1, u4, u3} D _inst_2 C _inst_1) (CategoryTheory.Functor.category.{u2, u1, u4, u3} D _inst_2 C _inst_1) G G' (CategoryTheory.Adjunction.rightAdjointUniq.{u1, u2, u3, u4} C _inst_1 D _inst_2 F G G' adj1 adj2)) x) (CategoryTheory.NatTrans.app.{u2, u1, u4, u3} D _inst_2 C _inst_1 G' G'' (CategoryTheory.Iso.hom.{max u4 u1, max (max (max u3 u4) u1) u2} (CategoryTheory.Functor.{u2, u1, u4, u3} D _inst_2 C _inst_1) (CategoryTheory.Functor.category.{u2, u1, u4, u3} D _inst_2 C _inst_1) G' G'' (CategoryTheory.Adjunction.rightAdjointUniq.{u1, u2, u3, u4} C _inst_1 D _inst_2 F G' G'' adj2 adj3)) x)) (CategoryTheory.NatTrans.app.{u2, u1, u4, u3} D _inst_2 C _inst_1 G G'' (CategoryTheory.Iso.hom.{max u4 u1, max (max (max u3 u4) u1) u2} (CategoryTheory.Functor.{u2, u1, u4, u3} D _inst_2 C _inst_1) (CategoryTheory.Functor.category.{u2, u1, u4, u3} D _inst_2 C _inst_1) G G'' (CategoryTheory.Adjunction.rightAdjointUniq.{u1, u2, u3, u4} C _inst_1 D _inst_2 F G G'' adj1 adj3)) x)
Case conversion may be inaccurate. Consider using '#align category_theory.adjunction.right_adjoint_uniq_trans_app CategoryTheory.Adjunction.rightAdjointUniq_trans_appₓ'. -/
@[simp, reassoc.1]
theorem rightAdjointUniq_trans_app {F : C ⥤ D} {G G' G'' : D ⥤ C} (adj1 : F ⊣ G) (adj2 : F ⊣ G')
    (adj3 : F ⊣ G'') (x : D) :
    (rightAdjointUniq adj1 adj2).Hom.app x ≫ (rightAdjointUniq adj2 adj3).Hom.app x =
      (rightAdjointUniq adj1 adj3).Hom.app x :=
  by
  apply Quiver.Hom.op_inj
  exact
    left_adjoint_uniq_trans_app (op_adjoint_op_of_adjoint _ _ adj3)
      (op_adjoint_op_of_adjoint _ _ adj2) (op_adjoint_op_of_adjoint _ _ adj1) (Opposite.op x)
#align category_theory.adjunction.right_adjoint_uniq_trans_app CategoryTheory.Adjunction.rightAdjointUniq_trans_app

#print CategoryTheory.Adjunction.rightAdjointUniq_trans /-
@[simp, reassoc.1]
theorem rightAdjointUniq_trans {F : C ⥤ D} {G G' G'' : D ⥤ C} (adj1 : F ⊣ G) (adj2 : F ⊣ G')
    (adj3 : F ⊣ G'') :
    (rightAdjointUniq adj1 adj2).Hom ≫ (rightAdjointUniq adj2 adj3).Hom =
      (rightAdjointUniq adj1 adj3).Hom :=
  by
  ext
  simp
#align category_theory.adjunction.right_adjoint_uniq_trans CategoryTheory.Adjunction.rightAdjointUniq_trans
-/

#print CategoryTheory.Adjunction.rightAdjointUniq_refl /-
@[simp]
theorem rightAdjointUniq_refl {F : C ⥤ D} {G : D ⥤ C} (adj1 : F ⊣ G) :
    (rightAdjointUniq adj1 adj1).Hom = 𝟙 _ :=
  by
  delta right_adjoint_uniq
  simp
#align category_theory.adjunction.right_adjoint_uniq_refl CategoryTheory.Adjunction.rightAdjointUniq_refl
-/

#print CategoryTheory.Adjunction.natIsoOfLeftAdjointNatIso /-
/-- Given two adjunctions, if the left adjoints are naturally isomorphic, then so are the right
adjoints.
-/
def natIsoOfLeftAdjointNatIso {F F' : C ⥤ D} {G G' : D ⥤ C} (adj1 : F ⊣ G) (adj2 : F' ⊣ G')
    (l : F ≅ F') : G ≅ G' :=
  rightAdjointUniq adj1 (adj2.ofNatIsoLeft l.symm)
#align category_theory.adjunction.nat_iso_of_left_adjoint_nat_iso CategoryTheory.Adjunction.natIsoOfLeftAdjointNatIso
-/

#print CategoryTheory.Adjunction.natIsoOfRightAdjointNatIso /-
/-- Given two adjunctions, if the right adjoints are naturally isomorphic, then so are the left
adjoints.
-/
def natIsoOfRightAdjointNatIso {F F' : C ⥤ D} {G G' : D ⥤ C} (adj1 : F ⊣ G) (adj2 : F' ⊣ G')
    (r : G ≅ G') : F ≅ F' :=
  leftAdjointUniq adj1 (adj2.ofNatIsoRight r.symm)
#align category_theory.adjunction.nat_iso_of_right_adjoint_nat_iso CategoryTheory.Adjunction.natIsoOfRightAdjointNatIso
-/

end CategoryTheory.Adjunction