-- 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))
x : (↑(↑X.toPresheafedSpace : TopCat) : Type v)
hx : x ∈ (Opens.map f).obj (unop (op U))
e_hom :
  Eq (α := CC (X.presheaf.stalk x))
    ((ConcreteCategory.hom (FC := FC) (CC := CC) ({ base := f, c := fc : X.Hom Y.toPresheafedSpace }.stalkMap x) :
          (a :
              CC
                (Y.presheaf.stalk
                  ((ConcreteCategory.hom { base := f, c := fc }.base :
                      (a : (↑(↑X.toPresheafedSpace : TopCat) : Type v)) → (↑(↑Y.toPresheafedSpace : TopCat) : Type v))
                    x))) →
            CC (X.presheaf.stalk x))
        ((ConcreteCategory.hom (FC := FC) (CC := CC)
                (Y.presheaf.germ (unop (op U))
                  ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC :=
                        carrier) { base := f, c := fc : X.Hom Y.toPresheafedSpace }.base :
                      (a : (↑(↑X.toPresheafedSpace : TopCat) : Type v)) → (↑(↑Y.toPresheafedSpace : TopCat) : Type v))
                    x)
                  hx) :
              (a : CC (Y.presheaf.obj (op (unop (op U))))) →
                CC
                  (Y.presheaf.stalk
                    ((ConcreteCategory.hom { base := f, c := fc }.base :
                        (a : (↑(↑X.toPresheafedSpace : TopCat) : Type v)) → (↑(↑Y.toPresheafedSpace : TopCat) : Type v))
                      x)))
            s :
          CC
            (Y.presheaf.stalk
              ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := carrier)
                    { base := f, c := fc : X.Hom Y.toPresheafedSpace }.base :
                  (a : (↑(↑X.toPresheafedSpace : TopCat) : Type v)) → (↑(↑Y.toPresheafedSpace : TopCat) : Type v))
                x))) :
      CC (X.presheaf.stalk x))
    ((ConcreteCategory.hom (FC := FC) (CC := CC) ({ base := f, c := gc : X.Hom Y.toPresheafedSpace }.stalkMap x) :
          (a :
              CC
                (Y.presheaf.stalk
                  ((ConcreteCategory.hom { base := f, c := gc }.base :
                      (a : (↑(↑X.toPresheafedSpace : TopCat) : Type v)) → (↑(↑Y.toPresheafedSpace : TopCat) : Type v))
                    x))) →
            CC (X.presheaf.stalk x))
        ((ConcreteCategory.hom (FC := FC) (CC := CC)
                (Y.presheaf.germ (unop (op U))
                  ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC :=
                        carrier) { base := f, c := gc : X.Hom Y.toPresheafedSpace }.base :
                      (a : (↑(↑X.toPresheafedSpace : TopCat) : Type v)) → (↑(↑Y.toPresheafedSpace : TopCat) : Type v))
                    x)
                  hx) :
              (a : CC (Y.presheaf.obj (op (unop (op U))))) →
                CC
                  (Y.presheaf.stalk
                    ((ConcreteCategory.hom { base := f, c := gc }.base :
                        (a : (↑(↑X.toPresheafedSpace : TopCat) : Type v)) → (↑(↑Y.toPresheafedSpace : TopCat) : Type v))
                      x)))
            s :
          CC
            (Y.presheaf.stalk
              ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := carrier)
                    { base := f, c := gc : X.Hom Y.toPresheafedSpace }.base :
                  (a : (↑(↑X.toPresheafedSpace : TopCat) : Type v)) → (↑(↑Y.toPresheafedSpace : TopCat) : Type v))
                x))) :
      CC (X.presheaf.stalk x))
⊢ 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))

Tactic:
erw [← PresheafedSpace.stalkMap_germ_apply ⟨f, fc⟩, ← PresheafedSpace.stalkMap_germ_apply ⟨f, gc⟩]

Premises used: AlgebraicGeometry.PresheafedSpace.Hom.mk, AlgebraicGeometry.PresheafedSpace.stalkMap_germ_apply