-- Lean 4 proof step from Mathlib.Geometry.RingedSpace.OpenImmersion (goal)
Goal:
C : Type u
inst : Category.{v, u} C
X Y Z : PresheafedSpace.{v, u, u_1} C
f : X ⟶ Z
hf : IsOpenImmersion f
g : Y ⟶ Z
U : (Opens (↑(↑Z : TopCat) : Type u_1))ᵒᵖ
⊢ Eq (α :=
    Z.presheaf.obj U ⟶
      (Functor.op (Opens.map (Hom.base (Y.ofRestrict ⋯ ≫ g))) ⋙
            (Y.restrict (U := pullback f.base g.base) ⋯).presheaf).obj
        U)
    (NatTrans.app
        (CategoryStruct.comp (Y :=
          Functor.obj
            (TopCat.Presheaf.pushforward C (Hom.base (pullbackConeOfLeftFst f g ≫ f)) :
              TopCat.Presheaf.{u_1, v, u} C (↑(Y.restrict (U := pullback f.base g.base) ⋯) : TopCat) ⥤
                TopCat.Presheaf.{u_1, v, u} C (↑Z : TopCat))
            (Y.restrict (U := pullback f.base g.base) ⋯).presheaf)
          (Hom.c (pullbackConeOfLeftFst f g ≫ f) :
            Z.presheaf ⟶
              Functor.obj
                (TopCat.Presheaf.pushforward C (Hom.base (pullbackConeOfLeftFst f g ≫ f)) :
                  TopCat.Presheaf.{u_1, v, u} C (↑(Y.restrict (U := pullback f.base g.base) ⋯) : TopCat) ⥤
                    TopCat.Presheaf.{u_1, v, u} C (↑Z : TopCat))
                (Y.restrict (U := pullback f.base g.base) ⋯).presheaf)
          (Functor.whiskerRight (eqToHom ⋯) (Y.restrict (U := pullback f.base g.base) ⋯).presheaf :
            Functor.op (Opens.map (Hom.base (pullbackConeOfLeftFst f g ≫ f))) ⋙
                (Y.restrict (U := pullback f.base g.base) ⋯).presheaf ⟶
              Functor.op (Opens.map (Hom.base (Y.ofRestrict ⋯ ≫ g))) ⋙
                (Y.restrict (U := pullback f.base g.base) ⋯).presheaf))
        U :
      Z.presheaf.obj U ⟶
        (Functor.op (Opens.map (Hom.base (Y.ofRestrict ⋯ ≫ g))) ⋙
              (Y.restrict (U := pullback f.base g.base) ⋯).presheaf).obj
          U)
    (NatTrans.app (Hom.c (Y.ofRestrict ⋯ ≫ g)) U :
      Z.presheaf.obj U ⟶
        (Functor.op (Opens.map (Hom.base (Y.ofRestrict ⋯ ≫ g))) ⋙
              (Y.restrict (U := pullback f.base g.base) ⋯).presheaf).obj
          U)
have refine_2 : ∀ (unop : Opens (↑(↑Z : TopCat) : Type u_1)),
  Eq (α :=
    Z.presheaf.obj (op unop) ⟶
      (Functor.op (Opens.map (Hom.base (Y.ofRestrict sorry ≫ g))) ⋙
            (Y.restrict (U := pullback f.base g.base) sorry).presheaf).obj
        (op unop))
    (NatTrans.app
        (CategoryStruct.comp (Y :=
          Functor.obj
            (TopCat.Presheaf.pushforward C (Hom.base (pullbackConeOfLeftFst f g ≫ f)) :
              TopCat.Presheaf.{u_1, v, u} C (↑(Y.restrict (U := pullback f.base g.base) sorry) : TopCat) ⥤
                TopCat.Presheaf.{u_1, v, u} C (↑Z : TopCat))
            (Y.restrict (U := pullback f.base g.base) sorry).presheaf)
          (Hom.c (pullbackConeOfLeftFst f g ≫ f) :
            Z.presheaf ⟶
              Functor.obj
                (TopCat.Presheaf.pushforward C (Hom.base (pullbackConeOfLeftFst f g ≫ f)) :
                  TopCat.Presheaf.{u_1, v, u} C (↑(Y.restrict (U := pullback f.base g.base) sorry) : TopCat) ⥤
                    TopCat.Presheaf.{u_1, v, u} C (↑Z : TopCat))
                (Y.restrict (U := pullback f.base g.base) sorry).presheaf)
          (Functor.whiskerRight (eqToHom sorry) (Y.restrict (U := pullback f.base g.base) sorry).presheaf :
            Functor.op (Opens.map (Hom.base (pullbackConeOfLeftFst f g ≫ f))) ⋙
                (Y.restrict (U := pullback f.base g.base) sorry).presheaf ⟶
              Functor.op (Opens.map (Hom.base (Y.ofRestrict sorry ≫ g))) ⋙
                (Y.restrict (U := pullback f.base g.base) sorry).presheaf))
        (op unop) :
      Z.presheaf.obj (op unop) ⟶
        (Functor.op (Opens.map (Hom.base (Y.ofRestrict sorry ≫ g))) ⋙
              (Y.restrict (U := pullback f.base g.base) sorry).presheaf).obj
          (op unop))
    (NatTrans.app (Hom.c (Y.ofRestrict sorry ≫ g)) (op unop) :
      Z.presheaf.obj (op unop) ⟶
        (Functor.op (Opens.map (Hom.base (Y.ofRestrict sorry ≫ g))) ⋙
              (Y.restrict (U := pullback f.base g.base) sorry).presheaf).obj
          (op unop))
Tactic: induction U
Cleared hypotheses: U