-- Module: Mathlib.Geometry.RingedSpace.SheafedSpace
Goal:
C : Type u
inst : Category.{v, u} C
FC : C → C → Type u_1
CC : C → Type v
inst_1 : (X Y : C) → FunLike.{u_1 + 1, v + 1, v + 1} (FC X Y) (CC X) (CC Y)
instCC : ConcreteCategory C FC (CC := CC)
inst_2 : HasColimits.{v, u} C
inst_3 : HasLimits.{v, u} C
inst_4 : PreservesLimits (CategoryTheory.forget C (FC := FC) (CC := CC))
inst_5 : PreservesFilteredColimits (CategoryTheory.forget C (FC := FC) (CC := CC))
inst_6 : ReflectsIsomorphisms (CategoryTheory.forget C (FC := FC) (CC := CC))
X Y : SheafedSpace.{u, v, v} C
g : X ⟶ Y
f : (↑X.toPresheafedSpace : TopCat) ⟶ (↑Y.toPresheafedSpace : TopCat)
fc : Y.presheaf ⟶ obj (pushforward C f) X.presheaf
h : PresheafedSpace.Hom.base (InducedCategory.Hom.hom { hom := { base := f, c := fc } }) = g.hom.base
h' :
  ∀ (x : (↑(↑X.toPresheafedSpace : TopCat) : Type v)),
    Eq (α :=
      Y.presheaf.stalk
          ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := carrier) (X :=
                (↑X.toPresheafedSpace : TopCat)) (Y := (↑Y.toPresheafedSpace : TopCat))
                (PresheafedSpace.Hom.base
                  (InducedCategory.Hom.hom
                    { hom := { base := f, c := fc : X.Hom Y.toPresheafedSpace } :
                      InducedCategory.Hom (D := PresheafedSpace.{v, u, v} C) (F := toPresheafedSpace) X Y })) :
              (a : (↑(↑X.toPresheafedSpace : TopCat) : Type v)) → (↑(↑Y.toPresheafedSpace : TopCat) : Type v))
            x) ⟶
        X.presheaf.stalk x)
      (PresheafedSpace.Hom.stalkMap (InducedCategory.Hom.hom { hom := { base := f, c := fc } }) x)
      (Iso.hom (Y.presheaf.stalkCongr ⋯) ≫ PresheafedSpace.Hom.stalkMap g.hom x)
obtain :
  ∀ (g : (↑X.toPresheafedSpace : TopCat) ⟶ (↑Y.toPresheafedSpace : TopCat))
    (gc : Y.presheaf ⟶ obj (pushforward C g) X.presheaf)
    (h :
      Eq.{v + 1} (α := (↑X.toPresheafedSpace : TopCat) ⟶ (↑Y.toPresheafedSpace : TopCat))
        (PresheafedSpace.Hom.base (InducedCategory.Hom.hom { hom := { base := f, c := fc } }))
        (PresheafedSpace.Hom.base (InducedCategory.Hom.hom { hom := { base := g, c := gc } })))
    (h' :
      ∀ (x : (↑(↑X.toPresheafedSpace : TopCat) : Type v)),
        Eq (α :=
          Y.presheaf.stalk
              ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := carrier)
                    (X := (↑X.toPresheafedSpace : TopCat)) (Y := (↑Y.toPresheafedSpace : TopCat))
                    (PresheafedSpace.Hom.base
                      (InducedCategory.Hom.hom
                        { hom := { base := f, c := fc : X.Hom Y.toPresheafedSpace } :
                          InducedCategory.Hom (D := PresheafedSpace.{v, u, v} C) (F := toPresheafedSpace) X Y })) :
                  (a : (↑(↑X.toPresheafedSpace : TopCat) : Type v)) → (↑(↑Y.toPresheafedSpace : TopCat) : Type v))
                x) ⟶
            X.presheaf.stalk x)
          (PresheafedSpace.Hom.stalkMap (InducedCategory.Hom.hom { hom := { base := f, c := fc } }) x)
          (CategoryStruct.comp (Y :=
            Y.presheaf.stalk
              ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := carrier)
                    (X := (↑X.toPresheafedSpace : TopCat)) (Y := (↑Y.toPresheafedSpace : TopCat))
                    (PresheafedSpace.Hom.base (InducedCategory.Hom.hom { hom := { base := g, c := gc } })) :
                  (a : (↑(↑X.toPresheafedSpace : TopCat) : Type v)) → (↑(↑Y.toPresheafedSpace : TopCat) : Type v))
                x))
            (Iso.hom (Y.presheaf.stalkCongr ⋯))
            (PresheafedSpace.Hom.stalkMap (InducedCategory.Hom.hom { hom := { base := g, c := gc } }) x))),
    Eq.{v + 1} (α := X ⟶ Y) { hom := { base := f, c := fc } } { hom := { base := g, c := gc } }
⊢ { hom := { base := f, c := fc } } = g

Tactic:
obtain ⟨g, gc⟩ := g