-- Lean 4 proof step from Mathlib.CategoryTheory.Subobject.Lattice (goal)
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
X Y Z : C
D : Type u₂
inst_1 : Category.{v₂, u₂} D
inst_2 : LocallySmall.{w, v₁, u₁} C
inst_3 : WellPowered.{w, v₁, u₁} C
inst_4 : HasWidePullbacks.{w, v₁, u₁} C
A : C
s : Set (Subobject.{v₁, u₁} A)
Z_1 : C
u v : Z_1 ⟶ widePullback.{w, v₁, u₁} s
h : u ≫ widePullbackι s = v ≫ widePullbackι s
val :
  (↑((⇑(equivShrink (Subobject.{v₁, u₁} A)) : (a : Subobject A) → Shrink.{w, max u₁ v₁} (Subobject A)) '' s) : Type w)
⊢ Eq (α :=
    Z_1 ⟶
      Functor.obj
        (wideCospan s :
          Functor.{w, v₁, w, u₁}
            (WidePullbackShape
              (↑((⇑(equivShrink (Subobject.{v₁, u₁} A)) : (a : Subobject A) → Shrink.{w, max u₁ v₁} (Subobject A)) ''
                    s) :
                Type w))
            C)
        (some val))
    (u ≫ limit.π (wideCospan s) (some val) :
      Z_1 ⟶
        Functor.obj
          (wideCospan s :
            Functor.{w, v₁, w, u₁}
              (WidePullbackShape
                (↑((⇑(equivShrink (Subobject.{v₁, u₁} A)) : (a : Subobject A) → Shrink.{w, max u₁ v₁} (Subobject A)) ''
                      s) :
                  Type w))
              C)
          (some val))
    (v ≫ limit.π (wideCospan s) (some val) :
      Z_1 ⟶
        Functor.obj
          (wideCospan s :
            Functor.{w, v₁, w, u₁}
              (WidePullbackShape
                (↑((⇑(equivShrink (Subobject.{v₁, u₁} A)) : (a : Subobject A) → Shrink.{w, max u₁ v₁} (Subobject A)) ''
                      s) :
                  Type w))
              C)
          (some val))
have some : Eq (α := Z_1 ⟶ A)
  (CategoryStruct.comp (Y :=
    Functor.obj (C := Subobject.{v₁, u₁} A) underlying
      ((Equiv.symm (equivShrink (Subobject.{v₁, u₁} A)) : (a : Shrink.{w, max u₁ v₁} (Subobject A)) → Subobject A)
        (↑val : Shrink.{w, max u₁ v₁} (Subobject A))))
    (u ≫ limit.π (wideCospan s) (some val) :
      Z_1 ⟶
        Functor.obj
          (wideCospan s :
            Functor.{w, v₁, w, u₁}
              (WidePullbackShape
                (↑((⇑(equivShrink (Subobject.{v₁, u₁} A)) : (a : Subobject A) → Shrink.{w, max u₁ v₁} (Subobject A)) ''
                      s) :
                  Type w))
              C)
          (some val))
    (arrow
        ((Equiv.symm (equivShrink (Subobject.{v₁, u₁} A)) : (a : Shrink.{w, max u₁ v₁} (Subobject A)) → Subobject A)
          (↑val : Shrink.{w, max u₁ v₁} (Subobject A))) :
      Functor.obj (C := Subobject.{v₁, u₁} A) underlying
          ((Equiv.symm (equivShrink (Subobject.{v₁, u₁} A)) : (a : Shrink.{w, max u₁ v₁} (Subobject A)) → Subobject A)
            (↑val : Shrink.{w, max u₁ v₁} (Subobject A))) ⟶
        A))
  (CategoryStruct.comp (Y :=
    Functor.obj (C := Subobject.{v₁, u₁} A) underlying
      ((Equiv.symm (equivShrink (Subobject.{v₁, u₁} A)) : (a : Shrink.{w, max u₁ v₁} (Subobject A)) → Subobject A)
        (↑val : Shrink.{w, max u₁ v₁} (Subobject A))))
    (v ≫ limit.π (wideCospan s) (some val) :
      Z_1 ⟶
        Functor.obj
          (wideCospan s :
            Functor.{w, v₁, w, u₁}
              (WidePullbackShape
                (↑((⇑(equivShrink (Subobject.{v₁, u₁} A)) : (a : Subobject A) → Shrink.{w, max u₁ v₁} (Subobject A)) ''
                      s) :
                  Type w))
              C)
          (some val))
    (arrow
        ((Equiv.symm (equivShrink (Subobject.{v₁, u₁} A)) : (a : Shrink.{w, max u₁ v₁} (Subobject A)) → Subobject A)
          (↑val : Shrink.{w, max u₁ v₁} (Subobject A))) :
      Functor.obj (C := Subobject.{v₁, u₁} A) underlying
          ((Equiv.symm (equivShrink (Subobject.{v₁, u₁} A)) : (a : Shrink.{w, max u₁ v₁} (Subobject A)) → Subobject A)
            (↑val : Shrink.{w, max u₁ v₁} (Subobject A))) ⟶
        A))
Tactic: apply (cancel_mono ((equivShrink (Subobject A)).symm _).arrow).1