-- 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
f : (↑X.toPresheafedSpace : TopCat) ⟶ (↑Y.toPresheafedSpace : TopCat)
fc gc : Y.presheaf ⟶ obj (pushforward C f) X.presheaf
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 := f, 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 := f, c := gc } }) x))
U : Opens (↑(↑Y.toPresheafedSpace : TopCat) : Type v)
s : CC (Y.presheaf.obj (op U))
e_hom :
  ∀ (x : (↑(↑X.toPresheafedSpace : TopCat) : Type v)) (hx : x ∈ (Opens.map f).obj (unop (op U))),
    Eq (α := CC (X.presheaf.stalk x))
      ((ConcreteCategory.hom (FC := FC) (CC := CC) (X.presheaf.germ ((Opens.map f).obj (unop (op U))) x hx) :
            (a : CC (X.presheaf.obj (op ((Opens.map f).obj (unop (op U)))))) → CC (X.presheaf.stalk x))
          ((ConcreteCategory.hom (FC := FC) (CC := CC) (fc.app (op U)) :
                (a : CC (Y.presheaf.obj (op U))) → CC (((pushforward C f).obj X.presheaf).obj (op U)))
              s :
            CC ((obj (pushforward C f) X.presheaf).obj (op U))) :
        CC (X.presheaf.stalk x))
      ((ConcreteCategory.hom (FC := FC) (CC := CC) (X.presheaf.germ ((Opens.map f).obj (unop (op U))) x hx) :
            (a : CC (X.presheaf.obj (op ((Opens.map f).obj (unop (op U)))))) → CC (X.presheaf.stalk x))
          ((ConcreteCategory.hom (FC := FC) (CC := CC) (gc.app (op U)) :
                (a : CC (Y.presheaf.obj (op U))) → CC (((pushforward C f).obj X.presheaf).obj (op U)))
              s :
            CC ((obj (pushforward C f) X.presheaf).obj (op U))) :
        CC (X.presheaf.stalk x))
⊢ Eq (α := CC ((obj (pushforward C f) X.presheaf).obj (op U)))
    ((ConcreteCategory.hom (FC := FC) (CC := CC) (fc.app (op U)) :
          (a : CC (Y.presheaf.obj (op U))) → CC (((pushforward C f).obj X.presheaf).obj (op U)))
        s :
      CC ((obj (pushforward C f) X.presheaf).obj (op U)))
    ((ConcreteCategory.hom (FC := FC) (CC := CC) (gc.app (op U)) :
          (a : CC (Y.presheaf.obj (op U))) → CC (((pushforward C f).obj X.presheaf).obj (op U)))
        s :
      CC ((obj (pushforward C f) X.presheaf).obj (op U)))

Tactic:
refine section_ext X.sheaf _ _ _ fun x hx ↦
    show X.presheaf.germ _ x _ _ = X.presheaf.germ _ x _ _ from ?_

Premises used: AlgebraicGeometry.SheafedSpace.sheaf, TopCat.Presheaf.section_ext