-- Lean 4 proof step from Mathlib.CategoryTheory.Subobject.Lattice (goal)
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
inst_1 : LocallySmall.{w, v₁, u₁} C
inst_2 : WellPowered.{w, v₁, u₁} C
inst_3 : HasWidePullbacks.{w, v₁, u₁} C
A : C
s : Set (Subobject.{v₁, u₁} A)
f : Subobject.{v₁, u₁} A
hf : f ∈ s
f₁ : Functor.obj underlying (sInf.{w, v₁, u₁} s) ⟶ Functor.obj underlying f
⊢ sInf.{w, v₁, u₁} s ≤ f
have w : (Iso.hom (underlyingIso (widePullbackι s)) ≫
      CategoryStruct.comp (Y :=
        Functor.obj.{w, v₁, w, u₁} (C :=
          WidePullbackShape
            (↑((⇑(equivShrink (Subobject.{v₁, u₁} A)) : (a : Subobject A) → Shrink.{w, max u₁ v₁} (Subobject A)) '' s) :
              Type w))
          (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
              (Subtype.mk (p := fun (x : Shrink.{w, max u₁ v₁} (Subobject.{v₁, u₁} A)) ↦
                Membership.mem (γ := Set (Shrink.{w, max u₁ v₁} (Subobject.{v₁, u₁} A)))
                  ((⇑(equivShrink (Subobject.{v₁, u₁} A)) : (a : Subobject A) → Shrink.{w, max u₁ v₁} (Subobject A)) ''
                    s)
                  x)
                ((equivShrink (Subobject.{v₁, u₁} A) : (a : Subobject A) → Shrink.{w, max u₁ v₁} (Subobject A)) f) sorry) :
            Option
              (↑((⇑(equivShrink (Subobject.{v₁, u₁} A)) : (a : Subobject A) → Shrink.{w, max u₁ v₁} (Subobject A)) ''
                    s) :
                Type w)))
        (limit.π (wideCospan s)
            (some
              (Subtype.mk (p := fun (x : Shrink.{w, max u₁ v₁} (Subobject.{v₁, u₁} A)) ↦
                Membership.mem (γ := Set (Shrink.{w, max u₁ v₁} (Subobject.{v₁, u₁} A)))
                  ((⇑(equivShrink (Subobject.{v₁, u₁} A)) : (a : Subobject A) → Shrink.{w, max u₁ v₁} (Subobject A)) ''
                    s)
                  x)
                ((equivShrink (Subobject.{v₁, u₁} A) : (a : Subobject A) → Shrink.{w, max u₁ v₁} (Subobject A)) f) sorry)) :
          limit.{w, w, v₁, u₁} (J :=
              WidePullbackShape
                (↑((⇑(equivShrink (Subobject.{v₁, u₁} A)) : (a : Subobject A) → Shrink.{w, max u₁ v₁} (Subobject A)) ''
                      s) :
                  Type w))
              (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) ⟶
            Functor.obj.{w, v₁, w, u₁} (C :=
              WidePullbackShape
                (↑((⇑(equivShrink (Subobject.{v₁, u₁} A)) : (a : Subobject A) → Shrink.{w, max u₁ v₁} (Subobject A)) ''
                      s) :
                  Type w))
              (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
                  (Subtype.mk (p := fun (x : Shrink.{w, max u₁ v₁} (Subobject.{v₁, u₁} A)) ↦
                    Membership.mem (γ := Set (Shrink.{w, max u₁ v₁} (Subobject.{v₁, u₁} A)))
                      ((⇑(equivShrink (Subobject.{v₁, u₁} A)) :
                          (a : Subobject A) → Shrink.{w, max u₁ v₁} (Subobject A)) ''
                        s)
                      x)
                    ((equivShrink (Subobject.{v₁, u₁} A) : (a : Subobject A) → Shrink.{w, max u₁ v₁} (Subobject A)) f)
                    sorry) :
                Option
                  (↑((⇑(equivShrink (Subobject.{v₁, u₁} A)) :
                          (a : Subobject A) → Shrink.{w, max u₁ v₁} (Subobject A)) ''
                        s) :
                    Type w)))
        (eqToHom sorry :
          Functor.obj.{w, v₁, w, u₁} (C :=
              WidePullbackShape
                (↑((⇑(equivShrink (Subobject.{v₁, u₁} A)) : (a : Subobject A) → Shrink.{w, max u₁ v₁} (Subobject A)) ''
                      s) :
                  Type w))
              (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
                  (Subtype.mk (p := fun (x : Shrink.{w, max u₁ v₁} (Subobject.{v₁, u₁} A)) ↦
                    Membership.mem (γ := Set (Shrink.{w, max u₁ v₁} (Subobject.{v₁, u₁} A)))
                      ((⇑(equivShrink (Subobject.{v₁, u₁} A)) :
                          (a : Subobject A) → Shrink.{w, max u₁ v₁} (Subobject A)) ''
                        s)
                      x)
                    ((equivShrink (Subobject.{v₁, u₁} A) : (a : Subobject A) → Shrink.{w, max u₁ v₁} (Subobject A)) f)
                    sorry) :
                Option
                  (↑((⇑(equivShrink (Subobject.{v₁, u₁} A)) :
                          (a : Subobject A) → Shrink.{w, max u₁ v₁} (Subobject A)) ''
                        s) :
                    Type w)) ⟶
            Functor.obj underlying f)) ≫
    f.arrow =
  (sInf.{w, v₁, u₁} s).arrow
Tactic: fapply le_of_comm