-- Lean 4 proof step from Mathlib.Topology.Sheaves.Functors (goal)
Goal:
C : Type u
inst : Category.{v, u} C
X Y : TopCat
f_1 : X ⟶ Y
ι : Type w
U : ι → Opens (↑Y : Type w)
A : Type u_1
inst_1 : Category.{w, u_1} A
f : X ⟶ Y
hf :
  IsOpenEmbedding (X := (↑X : Type w)) (Y := (↑Y : Type w))
    (⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{w}) ↦ C((↑X : Type w), (↑Y : Type w))) (CC := carrier) f) :
      (a : (↑X : Type w)) → (↑Y : Type w))
FA : A → A → Type u_2
CA : A → Type w
inst_2 : (X Y : A) → FunLike.{u_2 + 1, w + 1, w + 1} (FA X Y) (CA X) (CA Y)
inst_3 : ConcreteCategory A FA (CC := CA)
inst_4 : HasColimits.{w, u_1} A
inst_5 : HasLimits.{w, u_1} A
inst_6 : PreservesLimits (CategoryTheory.forget A (FC := FA) (CC := CA))
inst_7 : PreservesFilteredColimits (CategoryTheory.forget A (FC := FA) (CC := CA))
inst_8 : Functor.ReflectsIsomorphisms (CategoryTheory.forget A (FC := FA) (CC := CA))
X_1 Y_1 : TopCat.Sheaf A Y
u : X_1 ⟶ Y_1
⊢ Eq (α :=
    Functor.obj (presheafToSheaf (Opens.grothendieckTopology (↑X : Type w)) A)
        (Functor.obj (Presheaf.pullback A f) (Functor.obj (Sheaf.forget A Y) X_1)) ⟶
      Functor.obj (sheafPullback A hf) Y_1)
    (CategoryTheory.Functor.map (presheafToSheaf (Opens.grothendieckTopology (↑X : Type w)) A)
        (CategoryTheory.Functor.map (Presheaf.pullback A f) (CategoryTheory.Functor.map (Sheaf.forget A Y) u)) ≫
      CategoryTheory.Functor.map (presheafToSheaf (Opens.grothendieckTopology (↑X : Type w)) A)
          (NatTrans.app (Iso.hom ⋯.pullbackIso) (Functor.obj (Sheaf.forget A Y) Y_1)) ≫
        ObjectProperty.homMk (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type w)))
          (Iso.inv (isoSheafify (Opens.grothendieckTopology (↑X : Type w)) ⋯)))
    ((CategoryTheory.Functor.map (presheafToSheaf (Opens.grothendieckTopology (↑X : Type w)) A)
          (NatTrans.app (Iso.hom ⋯.pullbackIso) (Functor.obj (Sheaf.forget A Y) X_1)) ≫
        ObjectProperty.homMk (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type w)))
          (Iso.inv (isoSheafify (Opens.grothendieckTopology (↑X : Type w)) ⋯))) ≫
      CategoryTheory.Functor.map (sheafPullback A hf) u)
have refine_2 : Eq (α :=
  ObjectProperty.FullSubcategory.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type w)))
      (Functor.obj (presheafToSheaf (Opens.grothendieckTopology (↑X : Type w)) A)
        (Functor.obj (Presheaf.pullback A f) (Functor.obj (Sheaf.forget A Y) X_1))) ⟶
    (Functor.obj (sheafPullback A hf) Y_1).obj (P :=
      CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type w))))
  (InducedCategory.Hom.hom
    (CategoryTheory.Functor.map (presheafToSheaf (Opens.grothendieckTopology (↑X : Type w)) A)
        (NatTrans.app (Iso.hom sorry.pullbackIso) (Functor.obj (Sheaf.forget A Y) X_1) ≫
          CategoryTheory.Functor.map
            (Functor.obj (Functor.whiskeringLeft (Opens (↑X : Type w))ᵒᵖ (Opens (↑Y : Type w))ᵒᵖ A) sorry.functor.op)
            (CategoryTheory.Functor.map (Sheaf.forget A Y) u)) ≫
      ObjectProperty.homMk (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type w)))
        (Iso.inv (isoSheafify (Opens.grothendieckTopology (↑X : Type w)) sorry))))
  (InducedCategory.Hom.hom
    ((CategoryTheory.Functor.map (presheafToSheaf (Opens.grothendieckTopology (↑X : Type w)) A)
          (NatTrans.app (Iso.hom sorry.pullbackIso) (Functor.obj (Sheaf.forget A Y) X_1)) ≫
        ObjectProperty.homMk (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type w)))
          (Iso.inv (isoSheafify (Opens.grothendieckTopology (↑X : Type w)) sorry))) ≫
      CategoryTheory.Functor.map (sheafPullback A hf) u))
Tactic: rw [← Functor.map_comp_assoc, hf.isOpenMap.pullbackIso.hom.naturality, Sheaf.hom_ext_iff]