Module: Mathlib.CategoryTheory.Subobject.Comma

Goal state before:
pp: case refine_1
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
D : Type u₂
inst✝² : Category.{v₂, u₂} D
S : C ⥤ D
T : D
inst✝¹ : HasFiniteColimits C
inst✝ : PreservesFiniteColimits S
A : CostructuredArrow S T
P Q : (CostructuredArrow S T)ᵒᵖ
f : P ⟶ op A
g : Q ⟶ op A
hf : Mono f
hg : Mono g
h :
  { toFun := fun P ↦ ⟨projectQuotient P, ⋯⟩, invFun := fun P ↦ liftQuotient ↑P ⋯, left_inv := ⋯, right_inv := ⋯ }
      (Subobject.mk f) ≤
    { toFun := fun P ↦ ⟨projectQuotient P, ⋯⟩, invFun := fun P ↦ liftQuotient ↑P ⋯, left_inv := ⋯, right_inv := ⋯ }
      (Subobject.mk g)
⊢ P ⟶ Q

pp: case refine_2
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
D : Type u₂
inst✝² : Category.{v₂, u₂} D
S : C ⥤ D
T : D
inst✝¹ : HasFiniteColimits C
inst✝ : PreservesFiniteColimits S
A : CostructuredArrow S T
P Q : (CostructuredArrow S T)ᵒᵖ
f : P ⟶ op A
g : Q ⟶ op A
hf : Mono f
hg : Mono g
h :
  { toFun := fun P ↦ ⟨projectQuotient P, ⋯⟩, invFun := fun P ↦ liftQuotient ↑P ⋯, left_inv := ⋯, right_inv := ⋯ }
      (Subobject.mk f) ≤
    { toFun := fun P ↦ ⟨projectQuotient P, ⋯⟩, invFun := fun P ↦ liftQuotient ↑P ⋯, left_inv := ⋯, right_inv := ⋯ }
      (Subobject.mk g)
⊢ ?refine_1 ≫ g = f

pp: case refine_3
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
D : Type u₂
inst✝² : Category.{v₂, u₂} D
S : C ⥤ D
T : D
inst✝¹ : HasFiniteColimits C
inst✝ : PreservesFiniteColimits S
A : CostructuredArrow S T
P Q : (CostructuredArrow S T)ᵒᵖ
f : P ⟶ op A
g : Q ⟶ op A
hf : Mono f
hg : Mono g
h : Subobject.mk f ≤ Subobject.mk g
⊢ { toFun := fun P ↦ ⟨projectQuotient P, ⋯⟩, invFun := fun P ↦ liftQuotient ↑P ⋯, left_inv := ⋯, right_inv := ⋯ }
      (Subobject.mk f) ≤
    { toFun := fun P ↦ ⟨projectQuotient P, ⋯⟩, invFun := fun P ↦ liftQuotient ↑P ⋯, left_inv := ⋯, right_inv := ⋯ }
      (Subobject.mk g)

Tactic:
· refine (homMk (Subobject.ofMkLEMk _ _ h).unop ((cancel_epi (S.map g.unop.left)).1 ?_)).op
      dsimp
      simp only [← S.map_comp_assoc, unop_left_comp_ofMkLEMk_unop, unop_op, CommaMorphism.w,
        right_eq_id, Functor.const_obj_map]

Goal state after:
case refine_2
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
D : Type u₂
inst✝² : Category.{v₂, u₂} D
S : C ⥤ D
T : D
inst✝¹ : HasFiniteColimits C
inst✝ : PreservesFiniteColimits S
A : CostructuredArrow S T
P Q : (CostructuredArrow S T)ᵒᵖ
f : P ⟶ op A
g : Q ⟶ op A
hf : Mono f
hg : Mono g
h :
  { toFun := fun P ↦ ⟨projectQuotient P, ⋯⟩, invFun := fun P ↦ liftQuotient ↑P ⋯, left_inv := ⋯, right_inv := ⋯ }
      (Subobject.mk f) ≤
    { toFun := fun P ↦ ⟨projectQuotient P, ⋯⟩, invFun := fun P ↦ liftQuotient ↑P ⋯, left_inv := ⋯, right_inv := ⋯ }
      (Subobject.mk g)
⊢ (homMk (Subobject.ofMkLEMk (MonoOver.mk f).arrow.unop.left.op (MonoOver.mk g).arrow.unop.left.op h).unop ⋯).op ≫ g = f

case refine_3
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
D : Type u₂
inst✝² : Category.{v₂, u₂} D
S : C ⥤ D
T : D
inst✝¹ : HasFiniteColimits C
inst✝ : PreservesFiniteColimits S
A : CostructuredArrow S T
P Q : (CostructuredArrow S T)ᵒᵖ
f : P ⟶ op A
g : Q ⟶ op A
hf : Mono f
hg : Mono g
h : Subobject.mk f ≤ Subobject.mk g
⊢ { toFun := fun P ↦ ⟨projectQuotient P, ⋯⟩, invFun := fun P ↦ liftQuotient ↑P ⋯, left_inv := ⋯, right_inv := ⋯ }
      (Subobject.mk f) ≤
    { toFun := fun P ↦ ⟨projectQuotient P, ⋯⟩, invFun := fun P ↦ liftQuotient ↑P ⋯, left_inv := ⋯, right_inv := ⋯ }
      (Subobject.mk g)