-- 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
V₁ : Opens (↑X : Type v)
hxV₁ : x ∈ V₁
s₁ : ToType (FC := FC) (CC := CC) (F.presheaf.obj (op V₁))
hs₁ :
  Eq (α :=
    CC
      (obj (stalkFunctor C x)
        (G.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type v))))))
    ((ConcreteCategory.hom (FC := FC) (CC := ToType (FC := FC) (CC := CC))
            (germ (G.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type v)))) V₁ x hxV₁) :
          (a : ToType (G.obj.obj (op V₁))) → ToType (stalk G.obj x))
        ((ConcreteCategory.hom (FC := FC) (CC := ToType (FC := FC) (CC := CC)) (f.hom.app (op V₁)) :
              (a : ToType (F.obj.obj (op V₁))) → ToType (G.obj.obj (op V₁)))
            s₁ :
          ToType (FC := FC) (CC := CC)
            (G.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type v))).obj (op V₁))) :
      ToType (FC := FC) (CC := CC)
        (stalk (G.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type v)))) x))
    ((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))
V₂ : Opens (↑X : Type v)
hxV₂ : x ∈ V₂
iV₂V₁ : V₂ ⟶ V₁
iV₂U : V₂ ⟶ U
heq :
  Eq (α := CC (G.presheaf.obj (op V₂)))
    ((ConcreteCategory.hom (FC := FC) (CC := CC) (G.presheaf.map iV₂V₁.op) :
          (a : CC (G.presheaf.obj (op V₁))) → CC (G.presheaf.obj (op V₂)))
        ((ConcreteCategory.hom (FC := FC) (CC := ToType (FC := FC) (CC := CC)) (f.hom.app (op V₁)) :
              (a : ToType (F.obj.obj (op V₁))) → ToType (G.obj.obj (op V₁)))
            s₁ :
          ToType (FC := FC) (CC := CC)
            (G.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type v))).obj (op V₁))) :
      CC (G.presheaf.obj (op V₂)))
    ((ConcreteCategory.hom (FC := FC) (CC := CC) (G.presheaf.map iV₂U.op) :
          (a : CC (G.presheaf.obj (op U))) → CC (G.presheaf.obj (op V₂)))
        t :
      CC (G.presheaf.obj (op V₂)))
h₁ :
  Eq (α := CC (G.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type v))).obj (op V₂)))
    ((ConcreteCategory.hom (FC := FC) (CC := CC)
            (F.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type v))).map iV₂V₁.op ≫
              f.hom.app (op V₂)) :
          (a : CC (F.obj.obj (op V₁))) → CC (G.obj.obj (op V₂)))
        s₁ :
      CC (G.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type v))).obj (op V₂)))
    ((ConcreteCategory.hom (FC := FC) (CC := CC)
            (G.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type v))).map iV₂U.op) :
          (a : CC (G.obj.obj (op U))) → CC (G.obj.obj (op V₂)))
        t :
      CC (G.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type v))).obj (op V₂)))
h₂ :
  Eq (α := CC (G.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type v))).obj (op V₂)))
    ((ConcreteCategory.hom (FC := FC) (CC := CC)
            (f.hom.app (op V₁) ≫
              G.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type v))).map iV₂V₁.op) :
          (a : CC (F.obj.obj (op V₁))) → CC (G.obj.obj (op V₂)))
        s₁ :
      CC (G.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type v))).obj (op V₂)))
    ((ConcreteCategory.hom (FC := FC) (CC := CC)
            (G.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type v))).map iV₂U.op) :
          (a : CC (G.obj.obj (op U))) → CC (G.obj.obj (op V₂)))
        t :
      CC (G.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type v))).obj (op V₂)))
h₃ :
  Eq (α := CC (G.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type v))).obj (op V₂)))
    ((ConcreteCategory.hom (FC := FC) (CC := CC)
            (G.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type v))).map iV₂V₁.op) :
          (a : CC (G.obj.obj (op V₁))) → CC (G.obj.obj (op V₂)))
        ((ConcreteCategory.hom (FC := FC) (CC := CC) (f.hom.app (op V₁)) :
              (a : CC (F.obj.obj (op V₁))) → CC (G.obj.obj (op V₁)))
            s₁ :
          CC (G.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type v))).obj (op V₁))) :
      CC (G.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type v))).obj (op V₂)))
    ((ConcreteCategory.hom (FC := FC) (CC := CC)
            (G.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type v))).map iV₂U.op) :
          (a : CC (G.obj.obj (op U))) → CC (G.obj.obj (op V₂)))
        t :
      CC (G.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type v))).obj (op V₂)))
h₄ h₅ :
  Eq (α := CC (G.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type v))).obj (op V₂)))
    ((ConcreteCategory.hom (FC := FC) (CC := CC) (G.presheaf.map iV₂U.op) :
          (a : CC (G.presheaf.obj (op U))) → CC (G.presheaf.obj (op V₂)))
        t :
      CC (G.presheaf.obj (op V₂)))
    ((ConcreteCategory.hom (FC := FC) (CC := CC)
            (G.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type v))).map iV₂U.op) :
          (a : CC (G.obj.obj (op U))) → CC (G.obj.obj (op V₂)))
        t :
      CC (G.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type v))).obj (op V₂)))
⊢ Eq (α := CC (G.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type v))).obj (op V₂)))
    ((ConcreteCategory.hom (FC := FC) (CC := CC) (f.hom.app (op V₂)) :
          (a : CC (F.obj.obj (op V₂))) → CC (G.obj.obj (op V₂)))
        ((ConcreteCategory.hom (FC := FC) (CC := CC)
                (F.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type v))).map iV₂V₁.op) :
              (a : CC (F.obj.obj (op V₁))) → CC (F.obj.obj (op V₂)))
            s₁ :
          CC (F.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type v))).obj (op V₂))) :
      CC (G.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type v))).obj (op V₂)))
    ((ConcreteCategory.hom (FC := FC) (CC := CC)
            (G.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type v))).map iV₂U.op) :
          (a : CC (G.obj.obj (op U))) → CC (G.obj.obj (op V₂)))
        t :
      CC (G.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type v))).obj (op V₂)))

Tactic:
rw [← ConcreteCategory.comp_apply, f.1.naturality, ConcreteCategory.comp_apply, heq]

Premises used: CategoryTheory.ConcreteCategory.comp_apply, CategoryTheory.NatTrans.naturality, CategoryTheory.InducedCategory.Hom.hom