-- 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 : Sheaf C X
G : Presheaf C X
f : F.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type v))) ⟶ G
U : Opens (↑X : Type v)
h :
  ∀ x ∈ U,
    Function.Injective.{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))
      (⇑(ConcreteCategory.hom (FC := FC) (CC := CC) (map (stalkFunctor C x) f)) :
        (a : CC ((stalkFunctor C x).obj F.obj)) → CC ((stalkFunctor C x).obj G))
s t : CC (F.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type v))).obj (op U))
hst :
  Eq (α := CC (G.obj (op U)))
    ((ConcreteCategory.hom (FC := FC) (CC := CC) (f.app (op U)) : (a : CC (F.obj.obj (op U))) → CC (G.obj (op U))) s :
      CC (G.obj (op U)))
    ((ConcreteCategory.hom (FC := FC) (CC := CC) (f.app (op U)) : (a : CC (F.obj.obj (op U))) → CC (G.obj (op U))) t :
      CC (G.obj (op U)))
x : (↑X : Type v)
hx : x ∈ U
rw :
  Eq (α := CC (obj (stalkFunctor C x) G))
    ((ConcreteCategory.hom (FC := FC) (CC := CC) (G.germ U x hx) : (a : CC (G.obj (op U))) → CC (G.stalk x))
        ((ConcreteCategory.hom (FC := FC) (CC := CC) (f.app (op U)) : (a : CC (F.obj.obj (op U))) → CC (G.obj (op U)))
            s :
          CC (G.obj (op U))) :
      CC (G.stalk x))
    ((ConcreteCategory.hom (FC := FC) (CC := CC) (map (stalkFunctor C x) f) :
          (a : CC ((stalkFunctor C x).obj F.obj)) → CC ((stalkFunctor C x).obj G))
        ((ConcreteCategory.hom (FC := FC) (CC := CC) (F.presheaf.germ U x hx) :
              (a : CC (F.presheaf.obj (op U))) → CC (F.presheaf.stalk x))
            t :
          CC (F.presheaf.stalk x)) :
      CC (obj (stalkFunctor C x) G))
rw₁ :
  Eq (α := CC (obj (stalkFunctor C x) G))
    ((ConcreteCategory.hom (FC := FC) (CC := CC) (G.germ U x hx) : (a : CC (G.obj (op U))) → CC (G.stalk x))
        ((ConcreteCategory.hom (FC := FC) (CC := CC) (f.app (op U)) : (a : CC (F.obj.obj (op U))) → CC (G.obj (op U)))
            s :
          CC (G.obj (op U))) :
      CC (G.stalk x))
    ((ConcreteCategory.hom (FC := FC) (CC := CC) (G.germ U x hx) : (a : CC (G.obj (op U))) → CC (G.stalk x))
        ((ConcreteCategory.hom (FC := FC) (CC := CC) (f.app (op U)) : (a : CC (F.obj.obj (op U))) → CC (G.obj (op U)))
            t :
          CC (G.obj (op U))) :
      CC (G.stalk x))
rw₂ rw₃ :
  Eq (α := CC (obj (stalkFunctor C x) G))
    ((ConcreteCategory.hom (FC := FC) (CC := CC) (G.germ U x hx) : (a : CC (G.obj (op U))) → CC (G.stalk x))
        ((ConcreteCategory.hom (FC := FC) (CC := CC) (f.app (op U)) : (a : CC (F.obj.obj (op U))) → CC (G.obj (op U)))
            t :
          CC (G.obj (op U))) :
      CC (G.stalk x))
    ((ConcreteCategory.hom (FC := FC) (CC := CC) (G.germ U x hx) : (a : CC (G.obj (op U))) → CC (G.stalk x))
        ((ConcreteCategory.hom (FC := FC) (CC := CC) (f.app (op U)) : (a : CC (F.obj.obj (op U))) → CC (G.obj (op U)))
            t :
          CC (G.obj (op U))) :
      CC (G.stalk x))
⊢ Eq (α := CC (obj (stalkFunctor C x) G))
    ((ConcreteCategory.hom (FC := FC) (CC := CC) (map (stalkFunctor C x) f) :
          (a : CC ((stalkFunctor C x).obj F.obj)) → CC ((stalkFunctor C x).obj G))
        ((ConcreteCategory.hom (FC := FC) (CC := CC) (F.presheaf.germ U x hx) :
              (a : CC (F.presheaf.obj (op U))) → CC (F.presheaf.stalk x))
            s :
          CC (F.presheaf.stalk x)) :
      CC (obj (stalkFunctor C x) G))
    ((ConcreteCategory.hom (FC := FC) (CC := CC) (map (stalkFunctor C x) f) :
          (a : CC ((stalkFunctor C x).obj F.obj)) → CC ((stalkFunctor C x).obj G))
        ((ConcreteCategory.hom (FC := FC) (CC := CC) (F.presheaf.germ U x hx) :
              (a : CC (F.presheaf.obj (op U))) → CC (F.presheaf.stalk x))
            t :
          CC (F.presheaf.stalk x)) :
      CC (obj (stalkFunctor C x) G))

Tactic:
rw [stalkFunctor_map_germ_apply, stalkFunctor_map_germ_apply, hst]

Premises used: TopCat.Presheaf.stalkFunctor_map_germ_apply