/-- 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]
    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]
    erw [←
      colimit.w _
        (@homOfLE (OpenNhds x) _ ⟨_, U_property⟩
            ⟨(Opens.map f).obj (unop j_left), j_hom.unop.le U_property⟩ j_hom.unop.le).op]
    erw [colimit.ι_pre_assoc (Lan.diagram _ F _) (CostructuredArrow.map _)]
    erw [colimit.ι_pre_assoc (Lan.diagram _ F (op U_obj)) (CostructuredArrow.map _)]
    

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 }
⊢ colimit.ι (Lan.diagram (Opens.map f).op F (op U_obj))
        ((CostructuredArrow.map
              ((OpenNhds.inclusion x).op.map
                (homOfLE
                    (_ :
                      ((Functor.fromPUnit
                                ((OpenNhds.inclusion x).op.obj
                                  { unop := { obj := U_obj, property := U_property } })).obj
                            { as := PUnit.unit }).unop ≤
                        ((Opens.map f).op.obj j_left).unop)).op)).obj
          ((CostructuredArrow.map
                ((𝟙 (OpenNhds.map f x ⋙ OpenNhds.inclusion x).op).app
                  (op
                    { obj := j_left.unop,
                      property :=
                        (_ :
                          ↑{ val := f x,
                                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) } ∈
                            j_left.unop) }))).obj
            (CostructuredArrow.mk
              (𝟙
                (op
                  ((Opens.map f).obj
                    ((OpenNhds.inclusion (f x)).op.obj
                        (op
                          { obj := j_left.unop,
                            property :=
                              (_ :
                                ↑{ val := f x,
                                      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) } ∈
                                  j_left.unop) })).unop)))))) ≫
      colimit.ι
        (((whiskeringLeft (OpenNhds x)ᵒᵖ (Opens ↑X)ᵒᵖ C).obj (OpenNhds.inclusion x).op).obj
          ((lan (Opens.map f).op).obj F))
        (op { obj := U_obj, property := U_property }) =
    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:
rfl