-- Module: Mathlib.CategoryTheory.Subobject.Comma
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
D : Type u₂
inst_1 : Category.{v₂, u₂} D
S : Functor.{v₁, v₂, u₁, u₂} C D
T : D
inst_2 : HasFiniteColimits.{v₁, u₁} C
inst_3 : PreservesFiniteColimits S
A : CostructuredArrow S T
h :
  ∀ ⦃A_1 B : (CostructuredArrow S T)ᵒᵖ⦄ (f : A_1 ⟶ op A) (g : B ⟶ op A) [inst_4 : Mono f] [inst_5 : Mono g],
    LE.le (α :=
        { P : Subobject.{v₁, u₁} (op A.left) //
          ∃ (q : S.obj (unop (Functor.obj Subobject.underlying P)) ⟶ T), S.map P.arrow.unop ≫ q = A.hom })
        (({
                toFun := fun (P : Subobject (op A)) ↦
                  Subtype.mk (p := fun (P : Subobject.{v₁, u₁} (op A.left)) ↦
                    ∃ (q : S.obj (unop (Functor.obj Subobject.underlying P)) ⟶ T), S.map P.arrow.unop ≫ q = A.hom)
                    (projectQuotient P) ⋯,
                invFun :=
                  fun
                    (P :
                      { P : Subobject.{v₁, u₁} (op A.left) //
                        ∃ (q : S.obj (unop (Functor.obj Subobject.underlying P)) ⟶ T),
                          S.map P.arrow.unop ≫ q = A.hom }) ↦
                  liftQuotient (↑P : Subobject (op A.left)) ⋯,
                left_inv := ⋯, right_inv := ⋯ } :
              (a : Subobject (op A)) →
                { P : Subobject (op A.left) //
                  ∃ (q : S.obj (unop (Subobject.underlying.obj P)) ⟶ T), S.map P.arrow.unop ≫ q = A.hom })
            (Subobject.mk f) :
          { P : Subobject.{v₁, u₁} (op A.left) //
            ∃ (q : S.obj (unop (Functor.obj Subobject.underlying P)) ⟶ T), S.map P.arrow.unop ≫ q = A.hom })
        (({
                toFun := fun (P : Subobject (op A)) ↦
                  Subtype.mk (p := fun (P : Subobject.{v₁, u₁} (op A.left)) ↦
                    ∃ (q : S.obj (unop (Functor.obj Subobject.underlying P)) ⟶ T), S.map P.arrow.unop ≫ q = A.hom)
                    (projectQuotient P) ⋯,
                invFun :=
                  fun
                    (P :
                      { P : Subobject.{v₁, u₁} (op A.left) //
                        ∃ (q : S.obj (unop (Functor.obj Subobject.underlying P)) ⟶ T),
                          S.map P.arrow.unop ≫ q = A.hom }) ↦
                  liftQuotient (↑P : Subobject (op A.left)) ⋯,
                left_inv := ⋯, right_inv := ⋯ } :
              (a : Subobject (op A)) →
                { P : Subobject (op A.left) //
                  ∃ (q : S.obj (unop (Subobject.underlying.obj P)) ⟶ T), S.map P.arrow.unop ≫ q = A.hom })
            (Subobject.mk g) :
          { P : Subobject.{v₁, u₁} (op A.left) //
            ∃ (q : S.obj (unop (Functor.obj Subobject.underlying P)) ⟶ T), S.map P.arrow.unop ≫ q = A.hom }) ↔
      Subobject.mk f ≤ Subobject.mk g
⊢ ∀ {a b : Subobject (op A)},
    LE.le (α :=
        { P : Subobject.{v₁, u₁} (op A.left) //
          ∃ (q : S.obj (unop (Functor.obj Subobject.underlying P)) ⟶ T), S.map P.arrow.unop ≫ q = A.hom })
        (({
                toFun := fun (P : Subobject (op A)) ↦
                  Subtype.mk (p := fun (P : Subobject.{v₁, u₁} (op A.left)) ↦
                    ∃ (q : S.obj (unop (Functor.obj Subobject.underlying P)) ⟶ T), S.map P.arrow.unop ≫ q = A.hom)
                    (projectQuotient P) ⋯,
                invFun :=
                  fun
                    (P :
                      { P : Subobject.{v₁, u₁} (op A.left) //
                        ∃ (q : S.obj (unop (Functor.obj Subobject.underlying P)) ⟶ T),
                          S.map P.arrow.unop ≫ q = A.hom }) ↦
                  liftQuotient (↑P : Subobject (op A.left)) ⋯,
                left_inv := ⋯, right_inv := ⋯ } :
              (a : Subobject (op A)) →
                { P : Subobject (op A.left) //
                  ∃ (q : S.obj (unop (Subobject.underlying.obj P)) ⟶ T), S.map P.arrow.unop ≫ q = A.hom })
            a :
          { P : Subobject.{v₁, u₁} (op A.left) //
            ∃ (q : S.obj (unop (Functor.obj Subobject.underlying P)) ⟶ T), S.map P.arrow.unop ≫ q = A.hom })
        (({
                toFun := fun (P : Subobject (op A)) ↦
                  Subtype.mk (p := fun (P : Subobject.{v₁, u₁} (op A.left)) ↦
                    ∃ (q : S.obj (unop (Functor.obj Subobject.underlying P)) ⟶ T), S.map P.arrow.unop ≫ q = A.hom)
                    (projectQuotient P) ⋯,
                invFun :=
                  fun
                    (P :
                      { P : Subobject.{v₁, u₁} (op A.left) //
                        ∃ (q : S.obj (unop (Functor.obj Subobject.underlying P)) ⟶ T),
                          S.map P.arrow.unop ≫ q = A.hom }) ↦
                  liftQuotient (↑P : Subobject (op A.left)) ⋯,
                left_inv := ⋯, right_inv := ⋯ } :
              (a : Subobject (op A)) →
                { P : Subobject (op A.left) //
                  ∃ (q : S.obj (unop (Subobject.underlying.obj P)) ⟶ T), S.map P.arrow.unop ≫ q = A.hom })
            b :
          { P : Subobject.{v₁, u₁} (op A.left) //
            ∃ (q : S.obj (unop (Functor.obj Subobject.underlying P)) ⟶ T), S.map P.arrow.unop ≫ q = A.hom }) ↔
      a ≤ b

Tactic:
apply Subobject.ind₂

Premises used: CategoryTheory.Subobject.ind₂