-- Module: Mathlib.Topology.Sheaves.Stalks
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : HasColimits.{v, u} C
X : TopCat
FC : C → C → Type u_1
CC : C → Type v
inst_2 : (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_3 : PreservesFilteredColimits (forget C (FC := FC) (CC := CC))
inst_4 : HasLimits.{v, u} C
inst_5 : PreservesLimits (forget C (FC := FC) (CC := CC))
inst_6 : ReflectsIsomorphisms (forget C (FC := FC) (CC := CC))
F G : Sheaf C X
f : F ⟶ G
U : Opens (↑X : Type v)
h :
  ∀ x ∈ U,
    Function.Bijective.{v + 1, v + 1} (α :=
      CC
        (obj (stalkFunctor C x)
          (F.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type v))))))
      (β :=
      CC
        (obj (stalkFunctor C x)
          (G.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type v))))))
      (⇑(ConcreteCategory.hom (FC := FC) (CC := CC) (map (stalkFunctor C x) f.hom)) :
        (a : CC ((stalkFunctor C x).obj F.obj)) → CC ((stalkFunctor C x).obj G.obj))
t : CC (G.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type v))).obj (op U))
x : (↑X : Type v)
hx : x ∈ U
s₀ :
  CC (obj (stalkFunctor C x) (F.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type v)))))
hs₀ :
  Eq (α :=
    CC
      (obj (stalkFunctor C x)
        (G.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type v))))))
    ((ConcreteCategory.hom (FC := FC) (CC := CC) (map (stalkFunctor C x) f.hom) :
          (a : CC ((stalkFunctor C x).obj F.obj)) → CC ((stalkFunctor C x).obj G.obj))
        s₀ :
      CC
        (obj (stalkFunctor C x)
          (G.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type v))))))
    ((ConcreteCategory.hom (FC := FC) (CC := CC) (G.presheaf.germ U x hx) :
          (a : CC (G.presheaf.obj (op U))) → CC (G.presheaf.stalk x))
        t :
      CC (G.presheaf.stalk x))
obtain :
  ∀ (V₁ : Opens (↑X : Type v)) (hxV₁ : x ∈ V₁) (s₁ : ToType (FC := FC) (CC := CC) (F.presheaf.obj (op V₁)))
    (hs₀ :
      Eq (α :=
        CC
          (obj (stalkFunctor C x)
            (G.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type v))))))
        ((ConcreteCategory.hom (FC := FC) (CC := CC) (map (stalkFunctor C x) f.hom) :
              (a : CC ((stalkFunctor C x).obj F.obj)) → CC ((stalkFunctor C x).obj G.obj))
            ((ConcreteCategory.hom (FC := FC) (CC := CC) (F.presheaf.germ V₁ x hxV₁) :
                  (a : CC (F.presheaf.obj (op V₁))) → CC (F.presheaf.stalk x))
                s₁ :
              CC (F.presheaf.stalk x)) :
          CC
            (obj (stalkFunctor C x)
              (G.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type v))))))
        ((ConcreteCategory.hom (FC := FC) (CC := CC) (G.presheaf.germ U x hx) :
              (a : CC (G.presheaf.obj (op U))) → CC (G.presheaf.stalk x))
            t :
          CC (G.presheaf.stalk x))),
    ∃ (V : Opens (↑X : Type v)) (_ : x ∈ V) (iVU : V ⟶ U) (s :
      ToType (FC := FC) (CC := CC)
        (F.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type v))).obj (op V))),
      Eq (α := CC (G.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type v))).obj (op V)))
        ((ConcreteCategory.hom (FC := FC) (CC := CC) (f.hom.app (op V)) :
              (a : CC (F.obj.obj (op V))) → CC (G.obj.obj (op V)))
            s :
          CC (G.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type v))).obj (op V)))
        ((ConcreteCategory.hom (FC := FC) (CC := CC)
                (G.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type v))).map iVU.op) :
              (a : CC (G.obj.obj (op U))) → CC (G.obj.obj (op V)))
            t :
          CC (G.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type v))).obj (op V)))
⊢ ∃ (V : Opens (↑X : Type v)) (_ : x ∈ V) (iVU : V ⟶ U) (s :
    ToType (FC := FC) (CC := CC)
      (F.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type v))).obj (op V))),
    Eq (α := CC (G.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type v))).obj (op V)))
      ((ConcreteCategory.hom (FC := FC) (CC := CC) (f.hom.app (op V)) :
            (a : CC (F.obj.obj (op V))) → CC (G.obj.obj (op V)))
          s :
        CC (G.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type v))).obj (op V)))
      ((ConcreteCategory.hom (FC := FC) (CC := CC)
              (G.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type v))).map iVU.op) :
            (a : CC (G.obj.obj (op U))) → CC (G.obj.obj (op V)))
          t :
        CC (G.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type v))).obj (op V)))

Tactic:
obtain ⟨V₁, hxV₁, s₁, rfl⟩ := F.presheaf.exists_germ_eq s₀

Premises used: TopCat.Presheaf.exists_germ_eq, TopCat.Sheaf.presheaf, rfl