-- Lean 4 proof step from Mathlib.Geometry.RingedSpace.PresheafedSpace.Gluing (goal)
Goal:
C : Type u
inst : Category.{v, u} C
D : GlueData.{v, u} C
inst_1 : HasLimits.{v, u} C
i : D.J
U : Opens (↑(↑(D.U i) : TopCat) : Type v)
⊢ Eq (α :=
    (D.U i).presheaf.obj (op U) ⟶
      (D.U i).presheaf.obj
        (op
          (Functor.obj (C := Opens (↑(↑D.glued : TopCat) : Type v))
            (Opens.map (Hom.base (D.ι i)) :
              Opens (↑(↑D.glued : TopCat) : Type v) ⥤ Opens (↑(↑(D.U i) : TopCat) : Type v))
            (Functor.obj
                (IsOpenEmbedding.functor (f := Hom.base (D.ι i)) ⋯ :
                  Opens (↑(↑(D.U i) : TopCat) : Type v) ⥤ Opens (↑(↑D.glued : TopCat) : Type v))
                U :
              Opens (↑(↑D.glued : TopCat) : Type v)))))
    (D.opensImagePreimageMap i i U :
      (D.U i).presheaf.obj (op U) ⟶
        (D.U i).presheaf.obj
          (op
            (Functor.obj (C := Opens (↑(↑D.glued : TopCat) : Type v))
              (Opens.map (Hom.base (D.ι i)) :
                Opens (↑(↑D.glued : TopCat) : Type v) ⥤ Opens (↑(↑(D.U i) : TopCat) : Type v))
              (Functor.obj
                  (IsOpenEmbedding.functor (f := Hom.base (D.ι i)) ⋯ :
                    Opens (↑(↑(D.U i) : TopCat) : Type v) ⥤ Opens (↑(↑D.glued : TopCat) : Type v))
                  U :
                Opens (↑(↑D.glued : TopCat) : Type v)))))
    ((D.U i).presheaf.map (eqToHom ⋯) :
      (D.U i).presheaf.obj (op U) ⟶
        (D.U i).presheaf.obj
          (op
            (Functor.obj (C := Opens (↑(↑(Limits.colimit D.diagram.multispan) : TopCat) : Type v))
              (Opens.map (Hom.base (colimit.ι D.diagram.multispan (unop (op (WalkingMultispan.right i))))) :
                Opens (↑(↑(Limits.colimit D.diagram.multispan) : TopCat) : Type v) ⥤
                  Opens (↑(↑(D.diagram.multispan.obj (unop (op (WalkingMultispan.right i)))) : TopCat) : Type v))
              (Functor.obj
                  (IsOpenEmbedding.functor (f := Hom.base (D.ι i)) ⋯ :
                    Opens (↑(↑(D.U i) : TopCat) : Type v) ⥤ Opens (↑(↑D.glued : TopCat) : Type v))
                  U :
                Opens (↑(↑D.glued : TopCat) : Type v)))))
have h : Eq (α :=
  (D.U i).presheaf.obj (op U) ⟶
    (D.U i).presheaf.obj
      (op
        (Functor.obj (C := Opens (↑(↑D.glued : TopCat) : Type v))
          (Opens.map (Hom.base (D.ι i)) : Opens (↑(↑D.glued : TopCat) : Type v) ⥤ Opens (↑(↑(D.U i) : TopCat) : Type v))
          (Functor.obj
              (IsOpenEmbedding.functor (f := Hom.base (D.ι i)) sorry :
                Opens (↑(↑(D.U i) : TopCat) : Type v) ⥤ Opens (↑(↑D.glued : TopCat) : Type v))
              U :
            Opens (↑(↑D.glued : TopCat) : Type v)))))
  ((D.f i i).c.app (op U) ≫
      (D.t i i).c.app (op ((Opens.map (D.f i i).base).obj U)) ≫
        invApp (D.f i i) ((Opens.map (D.t i i).base).1 ((Opens.map (D.f i i).base).obj U)) ≫
          (D.U i).presheaf.map (Quiver.Hom.op (eqToHom sorry)) :
    (D.U i).presheaf.obj (op U) ⟶
      (D.U i).presheaf.obj
        (op
          (Functor.obj (C := Opens (↑(↑D.glued : TopCat) : Type v))
            (Opens.map (Hom.base (D.ι i)) :
              Opens (↑(↑D.glued : TopCat) : Type v) ⥤ Opens (↑(↑(D.U i) : TopCat) : Type v))
            (Functor.obj
                (IsOpenEmbedding.functor (f := Hom.base (D.ι i)) sorry :
                  Opens (↑(↑(D.U i) : TopCat) : Type v) ⥤ Opens (↑(↑D.glued : TopCat) : Type v))
                U :
              Opens (↑(↑D.glued : TopCat) : Type v)))))
  ((D.U i).presheaf.map (eqToHom sorry) :
    (D.U i).presheaf.obj (op U) ⟶
      (D.U i).presheaf.obj
        (op
          (Functor.obj (C := Opens (↑(↑(Limits.colimit D.diagram.multispan) : TopCat) : Type v))
            (Opens.map (Hom.base (Multicoequalizer.π D.diagram i)) :
              Opens (↑(↑(Limits.colimit D.diagram.multispan) : TopCat) : Type v) ⥤
                Opens (↑(↑(D.U i) : TopCat) : Type v))
            (Functor.obj
                (IsOpenEmbedding.functor (f := Hom.base (D.ι i)) sorry :
                  Opens (↑(↑(D.U i) : TopCat) : Type v) ⥤ Opens (↑(↑D.glued : TopCat) : Type v))
                U :
              Opens (↑(↑D.glued : TopCat) : Type v)))))
Tactic: dsimp [opensImagePreimageMap]