l3lab/ntp-mathlib huggingface dataset:
/-- 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]
    

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
        (Lan.diagram (Opens.map f).op F
          (op ((Opens.map f).obj ((OpenNhds.inclusion (f x)).op.obj (op { obj := obj✝, property := property✝ })).unop)))
        (CostructuredArrow.mk
          (𝟙
            (op
              ((Opens.map f).obj
                ((OpenNhds.inclusion (f x)).op.obj (op { obj := obj✝, property := property✝ })).unop)))) ≫
      colimit.pre
          (Lan.diagram (Opens.map f).op F
            ((OpenNhds.map f x ⋙ OpenNhds.inclusion x).op.obj (op { obj := obj✝, property := property✝ })))
          (CostructuredArrow.map
            ((𝟙 (OpenNhds.map f x ⋙ OpenNhds.inclusion x).op).app (op { obj := obj✝, property := property✝ }))) ≫
        colimit.ι
            (((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.obj (op { obj := obj✝, property := property✝ })) ≫
          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 x),
                      ι :=
                        NatTrans.mk fun V =>
                          colimit.ι ((OpenNhds.inclusion (f x)).op ⋙ F)
                            (op
                              { obj := V.left.unop,
                                property :=
                                  (_ :
                                    ↑{ val := f x,
                                          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:
erw [colimit.ι_desc, colimit.pre_desc, colimit.ι_desc, Category.comp_id]
/-- 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]
    

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✝
⊢ (Cocone.whisker
            (CostructuredArrow.map
              ((𝟙 (OpenNhds.map f x ⋙ OpenNhds.inclusion x).op).app (op { obj := obj✝, property := property✝ })))
            { pt := stalk F (f x),
              ι :=
                NatTrans.mk fun V =>
                  colimit.ι ((OpenNhds.inclusion (f x)).op ⋙ F)
                    (op
                      { obj := V.left.unop,
                        property :=
                          (_ :
                            ↑{ val := f x,
                                  property :=
                                    (_ :
                                      ↑{ val := x,
                                            property :=
                                              (_ :
                                                x ∈
                                                  ((OpenNhds.map f x).op.obj
                                                        (op { obj := obj✝, property := property✝ })).unop.obj) } ∈
                                        ↑((Opens.map f).op.obj V.left).unop) } ∈
                              V.left.unop) }) }).ι.app
      (CostructuredArrow.mk
        (𝟙
          (op
            ((Opens.map f).obj
              ((OpenNhds.inclusion (f x)).op.obj (op { obj := obj✝, property := property✝ })).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✝ })

Next tactic:
simp
/-- 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
    

Proof state:
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
⊢ stalkPullbackInv C f F x ≫ stalkPullbackHom C f F x = 𝟙 (stalk (pullbackObj f F) x)

Next tactic:
delta stalkPullbackHom stalkPullbackInv stalkFunctor Presheaf.pullback stalkPushforward
/-- 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
    

Proof state:
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
⊢ colimit.desc ((OpenNhds.inclusion x).op ⋙ pullbackObj f F)
        { pt := stalk F (f x),
          ι :=
            NatTrans.mk fun U => germToPullbackStalk C f F U.unop.obj { val := x, property := (_ : x ∈ U.unop.obj) } } ≫
      ((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 =
    𝟙 (stalk (pullbackObj f F) x)

Next tactic:
change (_ : colimit _ ⟶ _) = (_ : colimit _ ⟶ _)
/-- 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 _ ⟶ _)
    

Proof state:
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
⊢ colimit.desc ((OpenNhds.inclusion x).op ⋙ pullbackObj f F)
        { pt := stalk F (f x),
          ι :=
            NatTrans.mk fun U => germToPullbackStalk C f F U.unop.obj { val := x, property := (_ : x ∈ U.unop.obj) } } ≫
      ((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 =
    𝟙 (stalk (pullbackObj f F) x)

Next tactic:
ext ⟨U_obj, U_property⟩
/-- 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⟩
    

Proof state:
case w.mk.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
U_obj : Opens ↑X
U_property : x ∈ U_obj
⊢ 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 } } ≫
      colimit.desc ((OpenNhds.inclusion x).op ⋙ pullbackObj f F)
          { pt := stalk F (f x),
            ι :=
              NatTrans.mk fun U =>
                germToPullbackStalk C f F U.unop.obj { val := x, property := (_ : x ∈ U.unop.obj) } } ≫
        ((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.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 } } ≫
      𝟙 (stalk (pullbackObj f F) x)

Next tactic:
change (_ : colimit _ ⟶ _) = (_ : colimit _ ⟶ _)
/-- 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 _ ⟶ _)
    

Proof state:
case w.mk.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
U_obj : Opens ↑X
U_property : x ∈ U_obj
⊢ 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 } } ≫
      colimit.desc ((OpenNhds.inclusion x).op ⋙ pullbackObj f F)
          { pt := stalk F (f x),
            ι :=
              NatTrans.mk fun U =>
                germToPullbackStalk C f F U.unop.obj { val := x, property := (_ : x ∈ U.unop.obj) } } ≫
        ((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.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 } } ≫
      𝟙 (stalk (pullbackObj f F) x)

Next tactic:
ext ⟨j_left, ⟨⟨⟩⟩, j_hom⟩
/-- If `x` specializes to `y`, then there is a natural map `F.stalk y ⟶ F.stalk x`. -/
noncomputable def stalkSpecializes (F : X.Presheaf C) {x y : X} (h : x ⤳ y) :
    F.stalk y ⟶ F.stalk x := by
  

Proof state:
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasColimits C
X Y Z : TopCat
F : Presheaf C X
x y : ↑X
h : x ⤳ y
⊢ stalk F y ⟶ stalk F x

Next tactic:
refine' colimit.desc _ ⟨_, fun U => _, _⟩
/-- If `x` specializes to `y`, then there is a natural map `F.stalk y ⟶ F.stalk x`. -/
noncomputable def stalkSpecializes (F : X.Presheaf C) {x y : X} (h : x ⤳ y) :
    F.stalk y ⟶ F.stalk x := by
  refine' colimit.desc _ ⟨_, fun U => _, _⟩
  · 

Proof state:
case refine'_1
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasColimits C
X Y Z : TopCat
F : Presheaf C X
x y : ↑X
h : x ⤳ y
U : (OpenNhds y)ᵒᵖ
⊢ (((whiskeringLeft (OpenNhds y)ᵒᵖ (Opens ↑X)ᵒᵖ C).obj (OpenNhds.inclusion y).op).obj F).obj U ⟶
    ((Functor.const (OpenNhds y)ᵒᵖ).obj
          (colim.obj (((whiskeringLeft (OpenNhds x)ᵒᵖ (Opens ↑X)ᵒᵖ C).obj (OpenNhds.inclusion x).op).obj F))).obj
      U

Next tactic:
exact
      colimit.ι ((OpenNhds.inclusion x).op ⋙ F)
        (op ⟨(unop U).1, (specializes_iff_forall_open.mp h _ (unop U).1.2 (unop U).2 : _)⟩)
/-- If `x` specializes to `y`, then there is a natural map `F.stalk y ⟶ F.stalk x`. -/
noncomputable def stalkSpecializes (F : X.Presheaf C) {x y : X} (h : x ⤳ y) :
    F.stalk y ⟶ F.stalk x := by
  refine' colimit.desc _ ⟨_, fun U => _, _⟩
  · exact
      colimit.ι ((OpenNhds.inclusion x).op ⋙ F)
        (op ⟨(unop U).1, (specializes_iff_forall_open.mp h _ (unop U).1.2 (unop U).2 : _)⟩)
  · 

Proof state:
case refine'_2
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasColimits C
X Y Z : TopCat
F : Presheaf C X
x y : ↑X
h : x ⤳ y
⊢ ∀ ⦃X_1 Y : (OpenNhds y)ᵒᵖ⦄ (f : X_1 ⟶ Y),
    (((whiskeringLeft (OpenNhds y)ᵒᵖ (Opens ↑X)ᵒᵖ C).obj (OpenNhds.inclusion y).op).obj F).map f ≫
        (fun U =>
            colimit.ι ((OpenNhds.inclusion x).op ⋙ F)
              (op { obj := U.unop.obj, property := (_ : x ∈ U.unop.obj.carrier) }))
          Y =
      (fun U =>
            colimit.ι ((OpenNhds.inclusion x).op ⋙ F)
              (op { obj := U.unop.obj, property := (_ : x ∈ U.unop.obj.carrier) }))
          X_1 ≫
        ((Functor.const (OpenNhds y)ᵒᵖ).obj
              (colim.obj (((whiskeringLeft (OpenNhds x)ᵒᵖ (Opens ↑X)ᵒᵖ C).obj (OpenNhds.inclusion x).op).obj F))).map
          f

Next tactic:
intro U V i
/-- If `x` specializes to `y`, then there is a natural map `F.stalk y ⟶ F.stalk x`. -/
noncomputable def stalkSpecializes (F : X.Presheaf C) {x y : X} (h : x ⤳ y) :
    F.stalk y ⟶ F.stalk x := by
  refine' colimit.desc _ ⟨_, fun U => _, _⟩
  · exact
      colimit.ι ((OpenNhds.inclusion x).op ⋙ F)
        (op ⟨(unop U).1, (specializes_iff_forall_open.mp h _ (unop U).1.2 (unop U).2 : _)⟩)
  · intro U V i
    

Proof state:
case refine'_2
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasColimits C
X Y Z : TopCat
F : Presheaf C X
x y : ↑X
h : x ⤳ y
U V : (OpenNhds y)ᵒᵖ
i : U ⟶ V
⊢ (((whiskeringLeft (OpenNhds y)ᵒᵖ (Opens ↑X)ᵒᵖ C).obj (OpenNhds.inclusion y).op).obj F).map i ≫
      (fun U =>
          colimit.ι ((OpenNhds.inclusion x).op ⋙ F)
            (op { obj := U.unop.obj, property := (_ : x ∈ U.unop.obj.carrier) }))
        V =
    (fun U =>
          colimit.ι ((OpenNhds.inclusion x).op ⋙ F)
            (op { obj := U.unop.obj, property := (_ : x ∈ U.unop.obj.carrier) }))
        U ≫
      ((Functor.const (OpenNhds y)ᵒᵖ).obj
            (colim.obj (((whiskeringLeft (OpenNhds x)ᵒᵖ (Opens ↑X)ᵒᵖ C).obj (OpenNhds.inclusion x).op).obj F))).map
        i

Next tactic:
dsimp
/-- If `x` specializes to `y`, then there is a natural map `F.stalk y ⟶ F.stalk x`. -/
noncomputable def stalkSpecializes (F : X.Presheaf C) {x y : X} (h : x ⤳ y) :
    F.stalk y ⟶ F.stalk x := by
  refine' colimit.desc _ ⟨_, fun U => _, _⟩
  · exact
      colimit.ι ((OpenNhds.inclusion x).op ⋙ F)
        (op ⟨(unop U).1, (specializes_iff_forall_open.mp h _ (unop U).1.2 (unop U).2 : _)⟩)
  · intro U V i
    dsimp
    

Proof state:
case refine'_2
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasColimits C
X Y Z : TopCat
F : Presheaf C X
x y : ↑X
h : x ⤳ y
U V : (OpenNhds y)ᵒᵖ
i : U ⟶ V
⊢ F.map ((OpenNhds.inclusion y).map i.unop).op ≫
      colimit.ι ((OpenNhds.inclusion x).op ⋙ F) (op { obj := V.unop.obj, property := (_ : x ∈ ↑V.unop.obj) }) =
    colimit.ι ((OpenNhds.inclusion x).op ⋙ F) (op { obj := U.unop.obj, property := (_ : x ∈ ↑U.unop.obj) }) ≫
      𝟙 (colimit ((OpenNhds.inclusion x).op ⋙ F))

Next tactic:
rw [Category.comp_id]
/-- If `x` specializes to `y`, then there is a natural map `F.stalk y ⟶ F.stalk x`. -/
noncomputable def stalkSpecializes (F : X.Presheaf C) {x y : X} (h : x ⤳ y) :
    F.stalk y ⟶ F.stalk x := by
  refine' colimit.desc _ ⟨_, fun U => _, _⟩
  · exact
      colimit.ι ((OpenNhds.inclusion x).op ⋙ F)
        (op ⟨(unop U).1, (specializes_iff_forall_open.mp h _ (unop U).1.2 (unop U).2 : _)⟩)
  · intro U V i
    dsimp
    rw [Category.comp_id]
    

Proof state:
case refine'_2
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasColimits C
X Y Z : TopCat
F : Presheaf C X
x y : ↑X
h : x ⤳ y
U V : (OpenNhds y)ᵒᵖ
i : U ⟶ V
⊢ F.map ((OpenNhds.inclusion y).map i.unop).op ≫
      colimit.ι ((OpenNhds.inclusion x).op ⋙ F) (op { obj := V.unop.obj, property := (_ : x ∈ ↑V.unop.obj) }) =
    colimit.ι ((OpenNhds.inclusion x).op ⋙ F) (op { obj := U.unop.obj, property := (_ : x ∈ ↑U.unop.obj) })

Next tactic:
let U' : OpenNhds x := ⟨_, (specializes_iff_forall_open.mp h _ (unop U).1.2 (unop U).2 : _)⟩
/-- If `x` specializes to `y`, then there is a natural map `F.stalk y ⟶ F.stalk x`. -/
noncomputable def stalkSpecializes (F : X.Presheaf C) {x y : X} (h : x ⤳ y) :
    F.stalk y ⟶ F.stalk x := by
  refine' colimit.desc _ ⟨_, fun U => _, _⟩
  · exact
      colimit.ι ((OpenNhds.inclusion x).op ⋙ F)
        (op ⟨(unop U).1, (specializes_iff_forall_open.mp h _ (unop U).1.2 (unop U).2 : _)⟩)
  · intro U V i
    dsimp
    rw [Category.comp_id]
    let U' : OpenNhds x := ⟨_, (specializes_iff_forall_open.mp h _ (unop U).1.2 (unop U).2 : _)⟩
    

Proof state:
case refine'_2
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasColimits C
X Y Z : TopCat
F : Presheaf C X
x y : ↑X
h : x ⤳ y
U V : (OpenNhds y)ᵒᵖ
i : U ⟶ V
U' : OpenNhds x := { obj := U.unop.obj, property := (_ : x ∈ U.unop.obj.carrier) }
⊢ F.map ((OpenNhds.inclusion y).map i.unop).op ≫
      colimit.ι ((OpenNhds.inclusion x).op ⋙ F) (op { obj := V.unop.obj, property := (_ : x ∈ ↑V.unop.obj) }) =
    colimit.ι ((OpenNhds.inclusion x).op ⋙ F) (op { obj := U.unop.obj, property := (_ : x ∈ ↑U.unop.obj) })

Next tactic:
let V' : OpenNhds x := ⟨_, (specializes_iff_forall_open.mp h _ (unop V).1.2 (unop V).2 : _)⟩
/-- If `x` specializes to `y`, then there is a natural map `F.stalk y ⟶ F.stalk x`. -/
noncomputable def stalkSpecializes (F : X.Presheaf C) {x y : X} (h : x ⤳ y) :
    F.stalk y ⟶ F.stalk x := by
  refine' colimit.desc _ ⟨_, fun U => _, _⟩
  · exact
      colimit.ι ((OpenNhds.inclusion x).op ⋙ F)
        (op ⟨(unop U).1, (specializes_iff_forall_open.mp h _ (unop U).1.2 (unop U).2 : _)⟩)
  · intro U V i
    dsimp
    rw [Category.comp_id]
    let U' : OpenNhds x := ⟨_, (specializes_iff_forall_open.mp h _ (unop U).1.2 (unop U).2 : _)⟩
    let V' : OpenNhds x := ⟨_, (specializes_iff_forall_open.mp h _ (unop V).1.2 (unop V).2 : _)⟩
    

Proof state:
case refine'_2
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasColimits C
X Y Z : TopCat
F : Presheaf C X
x y : ↑X
h : x ⤳ y
U V : (OpenNhds y)ᵒᵖ
i : U ⟶ V
U' : OpenNhds x := { obj := U.unop.obj, property := (_ : x ∈ U.unop.obj.carrier) }
V' : OpenNhds x := { obj := V.unop.obj, property := (_ : x ∈ V.unop.obj.carrier) }
⊢ F.map ((OpenNhds.inclusion y).map i.unop).op ≫
      colimit.ι ((OpenNhds.inclusion x).op ⋙ F) (op { obj := V.unop.obj, property := (_ : x ∈ ↑V.unop.obj) }) =
    colimit.ι ((OpenNhds.inclusion x).op ⋙ F) (op { obj := U.unop.obj, property := (_ : x ∈ ↑U.unop.obj) })

Next tactic:
exact colimit.w ((OpenNhds.inclusion x).op ⋙ F) (show V' ⟶ U' from i.unop).op
@[simp]
theorem stalkSpecializes_refl {C : Type*} [Category C] [Limits.HasColimits C] {X : TopCat}
    (F : X.Presheaf C) (x : X) : F.stalkSpecializes (specializes_refl x) = 𝟙 _ := by
  

Proof state:
C✝ : Type u
inst✝³ : Category.{v, u} C✝
inst✝² : HasColimits C✝
X✝ Y Z : TopCat
C : Type u_1
inst✝¹ : Category.{u_2, u_1} C
inst✝ : HasColimits C
X : TopCat
F : Presheaf C X
x : ↑X
⊢ stalkSpecializes F (_ : x ⤳ x) = 𝟙 (stalk F x)

Next tactic:
ext