/-- 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
    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]
    erw [colimit.ι_desc, colimit.pre_desc, colimit.ι_desc, Category.comp_id]
    simp
  inv_hom_id := by
    delta stalkPullbackHom stalkPullbackInv stalkFunctor Presheaf.pullback stalkPushforward
    change (_ : colimit _ ⟶ _) = (_ : colimit _ ⟶ _)
    ext ⟨U_obj, U_property⟩
    change (_ : colimit _ ⟶ _) = (_ : colimit _ ⟶ _)
    ext ⟨j_left, ⟨⟨⟩⟩, j_hom⟩
    erw [colimit.map_desc, colimit.map_desc, colimit.ι_desc_assoc, colimit.ι_desc_assoc,
      colimit.ι_desc, Category.comp_id]
    

Proof state:
case w.mk.mk.w.mk.mk.unit
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
U_obj : Opens ↑X
U_property : x ∈ U_obj
j_left : (Opens ↑Y)ᵒᵖ
j_hom :
  (Opens.map f).op.obj j_left ⟶
    (Functor.fromPUnit ((OpenNhds.inclusion x).op.obj { unop := { obj := U_obj, property := U_property } })).obj
      { as := PUnit.unit }
⊢ ((Cocones.precompose
                (((whiskeringLeft (OpenNhds (f x))ᵒᵖ (Opens ↑Y)ᵒᵖ C).obj (OpenNhds.inclusion (f x)).op).map
                  ((pushforwardPullbackAdjunction C f).unit.app F))).obj
            ((Cocones.precompose
                  (whiskerRight (NatTrans.op (OpenNhds.inclusionMapIso f x).inv) ((lan (Opens.map f).op).obj F))).obj
              (Cocone.whisker (OpenNhds.map f x).op
                (colimit.cocone
                  (((whiskeringLeft (OpenNhds x)ᵒᵖ (Opens ↑X)ᵒᵖ C).obj (OpenNhds.inclusion x).op).obj
                    ((lan (Opens.map f).op).obj F)))))).ι.app
      (op
        { obj := { left := j_left, right := { as := PUnit.unit }, hom := j_hom }.left.unop,
          property :=
            (_ :
              ↑{
                    val :=
                      f
                        ↑{ val := x,
                            property := (_ : x ∈ { unop := { obj := U_obj, property := U_property } }.unop.obj) },
                    property :=
                      (_ :
                        ↑{ val := x,
                              property := (_ : x ∈ { unop := { obj := U_obj, property := U_property } }.unop.obj) } ∈
                          ↑((Opens.map f).op.obj
                                { left := j_left, right := { as := PUnit.unit }, hom := j_hom }.left).unop) } ∈
                { left := j_left, right := { as := PUnit.unit }, hom := j_hom }.left.unop) }) =
    colimit.ι
        (Lan.diagram (Opens.map f).op F
          ((OpenNhds.inclusion x).op.obj { unop := { obj := U_obj, property := U_property } }))
        { left := j_left, right := { as := PUnit.unit }, hom := j_hom } ≫
      colimit.ι (((whiskeringLeft (OpenNhds x)ᵒᵖ (Opens ↑X)ᵒᵖ C).obj (OpenNhds.inclusion x).op).obj (pullbackObj f F))
        { unop := { obj := U_obj, property := U_property } }

Next tactic:
simp only [Cocone.whisker_ι, colimit.cocone_ι, OpenNhds.inclusionMapIso_inv,
      Cocones.precompose_obj_ι, whiskerRight_app, whiskerLeft_app, NatTrans.comp_app,
      whiskeringLeft_obj_map, NatTrans.op_id, lan_obj_map,
      pushforwardPullbackAdjunction_unit_app_app]