-- Module: Mathlib.Geometry.RingedSpace.PresheafedSpace.Gluing
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)
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)) ⋯ :
                  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)))))
⊢ Eq.{v + 1} (α := (D.U i).presheaf.obj (op U) ⟶ (D.diagramOverOpen U).obj (op (WalkingMultispan.right i)))
    (NatTrans.app
        (Cone.π
          { pt := (D.U i).presheaf.obj (op U),
            π :=
              { app := fun (j : (WalkingMultispan (MultispanShape.prod D.J))ᵒᵖ) ↦ D.ιInvAppπApp U (unop j),
                naturality := ⋯ } })
        (op (WalkingMultispan.right i)) :
      Functor.obj.{v, v, v, u} (C := (WalkingMultispan (MultispanShape.prod D.J))ᵒᵖ)
          (Functor.obj (Functor.const (WalkingMultispan (MultispanShape.prod D.J))ᵒᵖ)
            (Cone.pt.{v, v, v, u} (J := (WalkingMultispan (MultispanShape.prod D.J))ᵒᵖ) (F := D.diagramOverOpen U)
              { pt := (D.U i).presheaf.obj (op U),
                π :=
                  { app := fun (j : (WalkingMultispan (MultispanShape.prod D.J))ᵒᵖ) ↦ D.ιInvAppπApp U (unop j),
                    naturality := ⋯ :
                    NatTrans
                      (Functor.obj (Functor.const (WalkingMultispan (MultispanShape.prod D.J))ᵒᵖ)
                        ((D.U i).presheaf.obj (op U)))
                      (D.diagramOverOpen U) } :
                Cone (D.diagramOverOpen U) }))
          (op (WalkingMultispan.right i)) ⟶
        (D.diagramOverOpen U).obj (op (WalkingMultispan.right i)))
    ((D.U i).presheaf.map (eqToHom ⋯))

Tactic:
change D.opensImagePreimageMap i i U = _

Premises used: AlgebraicGeometry.PresheafedSpace.GlueData.opensImagePreimageMap