/-- The isomorphism `ℱ_{f(x)} ≅ (f⁻¹ℱ)ₓ`. -/
def stalkPullbackIso (f : X ⟶ Y) (F : Y.Presheaf C) (x : X) :
    F.stalk (f x) ≅ (pullbackObj f F).stalk x where
  hom := stalkPullbackHom _ _ _ _
  inv := stalkPullbackInv _ _ _ _
  hom_inv_id := by
    delta
      stalkPullbackHom stalkPullbackInv stalkFunctor Presheaf.pullback stalkPushforward
      germToPullbackStalk germ
    change (_ : colimit _ ⟶ _) = (_ : colimit _ ⟶ _)
    ext j
    induction' j with j
    cases j
    

Proof state:
case w.h.mk
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasColimits C
X Y Z : TopCat
f : X ⟶ Y
F : Presheaf C Y
x : ↑X
obj✝ : Opens ↑Y
property✝ : f x ∈ obj✝
⊢ colimit.ι (((whiskeringLeft (OpenNhds (f x))ᵒᵖ (Opens ↑Y)ᵒᵖ C).obj (OpenNhds.inclusion (f x)).op).obj F)
        (op { obj := obj✝, property := property✝ }) ≫
      (((whiskeringLeft (OpenNhds (f x))ᵒᵖ (Opens ↑Y)ᵒᵖ C).obj (OpenNhds.inclusion (f x)).op ⋙ colim).map
            ((pushforwardPullbackAdjunction C f).unit.app F) ≫
          colim.map (whiskerRight (NatTrans.op (OpenNhds.inclusionMapIso f x).inv) ((lan (Opens.map f).op).obj F)) ≫
            colimit.pre
              (((whiskeringLeft (OpenNhds x)ᵒᵖ (Opens ↑X)ᵒᵖ C).obj (OpenNhds.inclusion x).op).obj
                ((lan (Opens.map f).op).obj F))
              (OpenNhds.map f x).op) ≫
        colimit.desc ((OpenNhds.inclusion x).op ⋙ pullbackObj f F)
          { pt := stalk F (f x),
            ι :=
              NatTrans.mk fun U =>
                colimit.desc (Lan.diagram (Opens.map f).op F (op U.unop.obj))
                  { pt := stalk F (f ↑{ val := x, property := (_ : x ∈ U.unop.obj) }),
                    ι :=
                      NatTrans.mk fun V =>
                        colimit.ι
                          ((OpenNhds.inclusion
                                ↑{ val := f ↑{ val := x, property := (_ : x ∈ U.unop.obj) },
                                    property :=
                                      (_ :
                                        ↑{ val := x, property := (_ : x ∈ U.unop.obj) } ∈
                                          ↑((Opens.map f).op.obj V.left).unop) }).op ⋙
                            F)
                          (op
                            { obj := V.left.unop,
                              property :=
                                (_ :
                                  ↑{ val := f ↑{ val := x, property := (_ : x ∈ U.unop.obj) },
                                        property :=
                                          (_ :
                                            ↑{ val := x, property := (_ : x ∈ U.unop.obj) } ∈
                                              ↑((Opens.map f).op.obj V.left).unop) } ∈
                                    V.left.unop) }) } } =
    colimit.ι (((whiskeringLeft (OpenNhds (f x))ᵒᵖ (Opens ↑Y)ᵒᵖ C).obj (OpenNhds.inclusion (f x)).op).obj F)
        (op { obj := obj✝, property := property✝ }) ≫
      𝟙 (stalk F (f x))

Next tactic:
simp only [TopologicalSpace.OpenNhds.inclusionMapIso_inv, whiskerRight_app, whiskerLeft_app,
      whiskeringLeft_obj_map, Functor.comp_map, colimit.ι_map_assoc, NatTrans.op_id, lan_obj_map,
      pushforwardPullbackAdjunction_unit_app_app, Category.assoc, colimit.ι_pre_assoc]