Theorem: CategoryTheory.rightAdjointOfCostructuredArrowTerminalsAux_symm_apply
File: Mathlib/CategoryTheory/Adjunction/Comma.lean

Proof steps:
state_before: C : Type u₁
D : Type u₂
inst✝² : Category.{v₁, u₁} C
inst✝¹ : Category.{v₂, u₂} D
G : D ⥤ C
inst✝ : ∀ (A : C), HasTerminal (CostructuredArrow G A)
B : D
A : C
⊢ Function.LeftInverse (fun g => G.map g ≫ (⊤_ CostructuredArrow G A).hom) fun g =>
    (terminal.from (CostructuredArrow.mk g)).left
state_after: no goals
tactic: aesop_cat
state_before: C : Type u₁
D : Type u₂
inst✝² : Category.{v₁, u₁} C
inst✝¹ : Category.{v₂, u₂} D
G : D ⥤ C
inst✝ : ∀ (A : C), HasTerminal (CostructuredArrow G A)
B : D
A : C
g : B ⟶ (⊤_ CostructuredArrow G A).left
⊢ (fun g => (terminal.from (CostructuredArrow.mk g)).left) ((fun g => G.map g ≫ (⊤_ CostructuredArrow G A).hom) g) = g
state_after: C : Type u₁
D : Type u₂
inst✝² : Category.{v₁, u₁} C
inst✝¹ : Category.{v₂, u₂} D
G : D ⥤ C
inst✝ : ∀ (A : C), HasTerminal (CostructuredArrow G A)
B : D
A : C
g : B ⟶ (⊤_ CostructuredArrow G A).left
B' : CostructuredArrow G A := CostructuredArrow.mk (G.map g ≫ (⊤_ CostructuredArrow G A).hom)
⊢ (fun g => (terminal.from (CostructuredArrow.mk g)).left) ((fun g => G.map g ≫ (⊤_ CostructuredArrow G A).hom) g) = g
tactic: let B' : CostructuredArrow G A :=
      CostructuredArrow.mk (G.map g ≫ (⊤_ CostructuredArrow G A).hom)
state_before: C : Type u₁
D : Type u₂
inst✝² : Category.{v₁, u₁} C
inst✝¹ : Category.{v₂, u₂} D
G : D ⥤ C
inst✝ : ∀ (A : C), HasTerminal (CostructuredArrow G A)
B : D
A : C
g : B ⟶ (⊤_ CostructuredArrow G A).left
B' : CostructuredArrow G A := CostructuredArrow.mk (G.map g ≫ (⊤_ CostructuredArrow G A).hom)
⊢ (fun g => (terminal.from (CostructuredArrow.mk g)).left) ((fun g => G.map g ≫ (⊤_ CostructuredArrow G A).hom) g) = g
state_after: C : Type u₁
D : Type u₂
inst✝² : Category.{v₁, u₁} C
inst✝¹ : Category.{v₂, u₂} D
G : D ⥤ C
inst✝ : ∀ (A : C), HasTerminal (CostructuredArrow G A)
B : D
A : C
g : B ⟶ (⊤_ CostructuredArrow G A).left
B' : CostructuredArrow G A := CostructuredArrow.mk (G.map g ≫ (⊤_ CostructuredArrow G A).hom)
g' : B' ⟶ ⊤_ CostructuredArrow G A := CostructuredArrow.homMk g ⋯
⊢ (fun g => (terminal.from (CostructuredArrow.mk g)).left) ((fun g => G.map g ≫ (⊤_ CostructuredArrow G A).hom) g) = g
tactic: let g' : B' ⟶ ⊤_ CostructuredArrow G A := CostructuredArrow.homMk g rfl
state_before: C : Type u₁
D : Type u₂
inst✝² : Category.{v₁, u₁} C
inst✝¹ : Category.{v₂, u₂} D
G : D ⥤ C
inst✝ : ∀ (A : C), HasTerminal (CostructuredArrow G A)
B : D
A : C
g : B ⟶ (⊤_ CostructuredArrow G A).left
B' : CostructuredArrow G A := CostructuredArrow.mk (G.map g ≫ (⊤_ CostructuredArrow G A).hom)
g' : B' ⟶ ⊤_ CostructuredArrow G A := CostructuredArrow.homMk g ⋯
⊢ (fun g => (terminal.from (CostructuredArrow.mk g)).left) ((fun g => G.map g ≫ (⊤_ CostructuredArrow G A).hom) g) = g
state_after: C : Type u₁
D : Type u₂
inst✝² : Category.{v₁, u₁} C
inst✝¹ : Category.{v₂, u₂} D
G : D ⥤ C
inst✝ : ∀ (A : C), HasTerminal (CostructuredArrow G A)
B : D
A : C
g : B ⟶ (⊤_ CostructuredArrow G A).left
B' : CostructuredArrow G A := CostructuredArrow.mk (G.map g ≫ (⊤_ CostructuredArrow G A).hom)
g' : B' ⟶ ⊤_ CostructuredArrow G A := CostructuredArrow.homMk g ⋯
this : terminal.from B' = g'
⊢ (fun g => (terminal.from (CostructuredArrow.mk g)).left) ((fun g => G.map g ≫ (⊤_ CostructuredArrow G A).hom) g) = g
tactic: have : terminal.from _ = g' := by aesop_cat
state_before: C : Type u₁
D : Type u₂
inst✝² : Category.{v₁, u₁} C
inst✝¹ : Category.{v₂, u₂} D
G : D ⥤ C
inst✝ : ∀ (A : C), HasTerminal (CostructuredArrow G A)
B : D
A : C
g : B ⟶ (⊤_ CostructuredArrow G A).left
B' : CostructuredArrow G A := CostructuredArrow.mk (G.map g ≫ (⊤_ CostructuredArrow G A).hom)
g' : B' ⟶ ⊤_ CostructuredArrow G A := CostructuredArrow.homMk g ⋯
this : terminal.from B' = g'
⊢ (fun g => (terminal.from (CostructuredArrow.mk g)).left) ((fun g => G.map g ≫ (⊤_ CostructuredArrow G A).hom) g) = g
state_after: C : Type u₁
D : Type u₂
inst✝² : Category.{v₁, u₁} C
inst✝¹ : Category.{v₂, u₂} D
G : D ⥤ C
inst✝ : ∀ (A : C), HasTerminal (CostructuredArrow G A)
B : D
A : C
g : B ⟶ (⊤_ CostructuredArrow G A).left
B' : CostructuredArrow G A := CostructuredArrow.mk (G.map g ≫ (⊤_ CostructuredArrow G A).hom)
g' : B' ⟶ ⊤_ CostructuredArrow G A := CostructuredArrow.homMk g ⋯
this : terminal.from B' = g'
⊢ (terminal.from B').left = g
tactic: change CommaMorphism.left (terminal.from B') = _
state_before: C : Type u₁
D : Type u₂
inst✝² : Category.{v₁, u₁} C
inst✝¹ : Category.{v₂, u₂} D
G : D ⥤ C
inst✝ : ∀ (A : C), HasTerminal (CostructuredArrow G A)
B : D
A : C
g : B ⟶ (⊤_ CostructuredArrow G A).left
B' : CostructuredArrow G A := CostructuredArrow.mk (G.map g ≫ (⊤_ CostructuredArrow G A).hom)
g' : B' ⟶ ⊤_ CostructuredArrow G A := CostructuredArrow.homMk g ⋯
this : terminal.from B' = g'
⊢ (terminal.from B').left = g
state_after: C : Type u₁
D : Type u₂
inst✝² : Category.{v₁, u₁} C
inst✝¹ : Category.{v₂, u₂} D
G : D ⥤ C
inst✝ : ∀ (A : C), HasTerminal (CostructuredArrow G A)
B : D
A : C
g : B ⟶ (⊤_ CostructuredArrow G A).left
B' : CostructuredArrow G A := CostructuredArrow.mk (G.map g ≫ (⊤_ CostructuredArrow G A).hom)
g' : B' ⟶ ⊤_ CostructuredArrow G A := CostructuredArrow.homMk g ⋯
this : terminal.from B' = g'
⊢ g'.left = g
tactic: rw [this]
state_before: C : Type u₁
D : Type u₂
inst✝² : Category.{v₁, u₁} C
inst✝¹ : Category.{v₂, u₂} D
G : D ⥤ C
inst✝ : ∀ (A : C), HasTerminal (CostructuredArrow G A)
B : D
A : C
g : B ⟶ (⊤_ CostructuredArrow G A).left
B' : CostructuredArrow G A := CostructuredArrow.mk (G.map g ≫ (⊤_ CostructuredArrow G A).hom)
g' : B' ⟶ ⊤_ CostructuredArrow G A := CostructuredArrow.homMk g ⋯
this : terminal.from B' = g'
⊢ g'.left = g
state_after: no goals
tactic: rfl