-- Module: Mathlib.Topology.Sheaves.Functors
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
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)
            (X_1.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑Y : Type w)))))) ⟶
      hf.functor.op ⋙ Y_1.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑Y : Type w))))
    (InducedCategory.Hom.hom
        (CategoryTheory.Functor.map (presheafToSheaf (Opens.grothendieckTopology (↑X : Type w)) A)
          (NatTrans.app (Iso.hom ⋯.pullbackIso)
            (X_1.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑Y : Type w)))))) ≫
      InducedCategory.Hom.hom
          (CategoryTheory.Functor.map (presheafToSheaf (Opens.grothendieckTopology (↑X : Type w)) A)
            (⋯.functor.op.whiskerLeft u.hom)) ≫
        sheafifyLift (Opens.grothendieckTopology (↑X : Type w))
          (𝟙
            (hf.functor.op ⋙ Y_1.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑Y : Type w)))))
          ⋯)
    (InducedCategory.Hom.hom
        (CategoryTheory.Functor.map (presheafToSheaf (Opens.grothendieckTopology (↑X : Type w)) A)
          (NatTrans.app (Iso.hom ⋯.pullbackIso)
            (X_1.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑Y : Type w)))))) ≫
      sheafifyLift (Opens.grothendieckTopology (↑X : Type w))
          (𝟙
            (hf.functor.op ⋙ X_1.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑Y : Type w)))))
          ⋯ ≫
        hf.functor.op.whiskerLeft u.hom)
⊢ 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 ⋯.pullbackIso) (Functor.obj (Sheaf.forget A Y) X_1))) ≫
      InducedCategory.Hom.hom
          (CategoryTheory.Functor.map (presheafToSheaf (Opens.grothendieckTopology (↑X : Type w)) A)
            (⋯.functor.op.whiskerLeft (CategoryTheory.Functor.map (Sheaf.forget A Y) u))) ≫
        InducedCategory.Hom.hom
          (ObjectProperty.homMk (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type w)))
            (sheafifyLift (Opens.grothendieckTopology (↑X : Type w))
              (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 ⋯.pullbackIso) (Functor.obj (Sheaf.forget A Y) X_1))) ≫
      InducedCategory.Hom.hom
          (ObjectProperty.homMk (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type w)))
            (sheafifyLift (Opens.grothendieckTopology (↑X : Type w))
              (𝟙
                ((Functor.obj (sheafPullback A hf) X_1).obj (P :=
                  CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type w)))))
              ⋯)) ≫
        InducedCategory.Hom.hom (CategoryTheory.Functor.map (sheafPullback A hf) u))

Tactic:
dsimp [sheafPullback, Functor.sheafPushforwardContinuous, Sheaf.forget]

Premises used: Topology.IsOpenEmbedding.sheafPullback, CategoryTheory.Functor.sheafPushforwardContinuous, TopCat.Sheaf.forget