-- Module: Mathlib.CategoryTheory.Subobject.Lattice
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)
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) (Option.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)
            (Option.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) (Option.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)
            (Option.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))
⊢ 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)
        (Option.some val))
    (u ≫ limit.π (wideCospan s) (Option.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)
          (Option.some val))
    (v ≫ limit.π (wideCospan s) (Option.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)
          (Option.some val))

Tactic:
apply (cancel_mono ((equivShrink (Subobject A)).symm _).arrow).1

Premises used: CategoryTheory.Subobject.arrow, CategoryTheory.cancel_mono, Iff.mp, Equiv.symm, equivShrink, DFunLike.coe, CategoryTheory.Subobject