Theorem: CategoryTheory.Functor.leftOpRightOpEquiv_inverse_map
File: Mathlib/CategoryTheory/Opposites.lean

Proof steps:
state_before: C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
D : Type u₂
inst✝ : Category.{v₂, u₂} D
⊢ ∀ {X Y : (Cᵒᵖ ⥤ D)ᵒᵖ} (f : X ⟶ Y),
    (𝟭 (Cᵒᵖ ⥤ D)ᵒᵖ).map f ≫ ((fun F => (unop F).rightOpLeftOpIso.op) Y).hom =
      ((fun F => (unop F).rightOpLeftOpIso.op) X).hom ≫
        ({ obj := fun F => (unop F).rightOp, map := fun {X Y} η => NatTrans.rightOp η.unop, map_id := ⋯,
                map_comp := ⋯ } ⋙
              { obj := fun F => op F.leftOp, map := fun {X Y} η => (NatTrans.leftOp η).op, map_id := ⋯,
                map_comp := ⋯ }).map
          f
state_after: C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
D : Type u₂
inst✝ : Category.{v₂, u₂} D
F G : (Cᵒᵖ ⥤ D)ᵒᵖ
η : F ⟶ G
⊢ (𝟭 (Cᵒᵖ ⥤ D)ᵒᵖ).map η ≫ ((fun F => (unop F).rightOpLeftOpIso.op) G).hom =
    ((fun F => (unop F).rightOpLeftOpIso.op) F).hom ≫
      ({ obj := fun F => (unop F).rightOp, map := fun {X Y} η => NatTrans.rightOp η.unop, map_id := ⋯, map_comp := ⋯ } ⋙
            { obj := fun F => op F.leftOp, map := fun {X Y} η => (NatTrans.leftOp η).op, map_id := ⋯,
              map_comp := ⋯ }).map
        η
tactic: intro F G η
state_before: C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
D : Type u₂
inst✝ : Category.{v₂, u₂} D
F G : (Cᵒᵖ ⥤ D)ᵒᵖ
η : F ⟶ G
⊢ (𝟭 (Cᵒᵖ ⥤ D)ᵒᵖ).map η ≫ ((fun F => (unop F).rightOpLeftOpIso.op) G).hom =
    ((fun F => (unop F).rightOpLeftOpIso.op) F).hom ≫
      ({ obj := fun F => (unop F).rightOp, map := fun {X Y} η => NatTrans.rightOp η.unop, map_id := ⋯, map_comp := ⋯ } ⋙
            { obj := fun F => op F.leftOp, map := fun {X Y} η => (NatTrans.leftOp η).op, map_id := ⋯,
              map_comp := ⋯ }).map
        η
state_after: C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
D : Type u₂
inst✝ : Category.{v₂, u₂} D
F G : (Cᵒᵖ ⥤ D)ᵒᵖ
η : F ⟶ G
⊢ η ≫ (unop G).rightOpLeftOpIso.hom.op =
    (unop F).rightOpLeftOpIso.hom.op ≫ (NatTrans.leftOp (NatTrans.rightOp η.unop)).op
tactic: dsimp
state_before: C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
D : Type u₂
inst✝ : Category.{v₂, u₂} D
F G : (Cᵒᵖ ⥤ D)ᵒᵖ
η : F ⟶ G
⊢ η ≫ (unop G).rightOpLeftOpIso.hom.op =
    (unop F).rightOpLeftOpIso.hom.op ≫ (NatTrans.leftOp (NatTrans.rightOp η.unop)).op
state_after: C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
D : Type u₂
inst✝ : Category.{v₂, u₂} D
F G : (Cᵒᵖ ⥤ D)ᵒᵖ
η : F ⟶ G
⊢ ((unop G).rightOpLeftOpIso.hom ≫ η.unop).op =
    (NatTrans.leftOp (NatTrans.rightOp η.unop.op.unop) ≫ (unop F).rightOpLeftOpIso.hom).op
tactic: rw [show η = η.unop.op by simp, ← op_comp, ← op_comp]
state_before: C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
D : Type u₂
inst✝ : Category.{v₂, u₂} D
F G : (Cᵒᵖ ⥤ D)ᵒᵖ
η : F ⟶ G
⊢ ((unop G).rightOpLeftOpIso.hom ≫ η.unop).op =
    (NatTrans.leftOp (NatTrans.rightOp η.unop.op.unop) ≫ (unop F).rightOpLeftOpIso.hom).op
state_after: case e_f
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
D : Type u₂
inst✝ : Category.{v₂, u₂} D
F G : (Cᵒᵖ ⥤ D)ᵒᵖ
η : F ⟶ G
⊢ (unop G).rightOpLeftOpIso.hom ≫ η.unop =
    NatTrans.leftOp (NatTrans.rightOp η.unop.op.unop) ≫ (unop F).rightOpLeftOpIso.hom
tactic: congr 1
state_before: case e_f
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
D : Type u₂
inst✝ : Category.{v₂, u₂} D
F G : (Cᵒᵖ ⥤ D)ᵒᵖ
η : F ⟶ G
⊢ (unop G).rightOpLeftOpIso.hom ≫ η.unop =
    NatTrans.leftOp (NatTrans.rightOp η.unop.op.unop) ≫ (unop F).rightOpLeftOpIso.hom
state_after: no goals
tactic: aesop_cat