awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Module: Mathlib.Topology.Sheaves.Stalks
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : HasColimits.{v, u} C
X Y Z : TopCat
FC : C → C → Type u_1
CC : C → Type u_2
inst_2 : (X Y : C) → FunLike.{u_1 + 1, u_2 + 1, u_2 + 1} (FC X Y) (CC X) (CC Y)
f : X ⟶ Y
F : Presheaf C Y
x : (↑X : Type v)
ih :
  ∀ (V : Opens (↑X : Type v)) (hV : x ∈ V) (V_1 : Opens (↑Y : Type v)) (hV_1 : unop (op V) ≤ (Opens.map f).obj V_1),
    NatTrans.app (NatTrans.app (Adjunction.unit (pullbackPushforwardAdjunction C f)) F) (op V_1) ≫
        (obj (pullback C f) F).map (homOfLE hV_1).op ≫
          (obj (pullback C f) F).germ V x hV ≫ stalkPullbackInv C f F x ≫ stalkPullbackHom C f F x =
      NatTrans.app (NatTrans.app (Adjunction.unit (pullbackPushforwardAdjunction C f)) F) (op V_1) ≫
        (obj (pullback C f) F).map (homOfLE hV_1).op ≫
          (obj (pullback C f) F).germ V x hV ≫ 𝟙 ((obj (pullback C f) F).stalk x)
⊢ stalkPullbackInv C f F x ≫ stalkPullbackHom C f F x = 𝟙 ((obj (pullback C f) F).stalk x)

Tactic:
ext V hV
-- Module: Mathlib.Topology.Sheaves.Stalks
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : HasColimits.{v, u} C
X Y Z : TopCat
FC : C → C → Type u_1
CC : C → Type u_2
inst_2 : (X Y : C) → FunLike.{u_1 + 1, u_2 + 1, u_2 + 1} (FC X Y) (CC X) (CC Y)
f : X ⟶ Y
F : Presheaf C Y
x : (↑X : Type v)
V : Opens (↑X : Type v)
hV : x ∈ V
V_1 : Opens (↑Y : Type v)
hV_1 : unop (op V) ≤ (Opens.map f).obj V_1
ih :
  NatTrans.app (NatTrans.app (Adjunction.unit (pullbackPushforwardAdjunction C f)) F) (op V_1) ≫
      (obj (pullback C f) F).map (homOfLE hV_1).op ≫
        (obj (pullback C f) F).germ V x hV ≫ stalkPullbackInv C f F x ≫ stalkPullbackHom C f F x =
    NatTrans.app (NatTrans.app (Adjunction.unit (pullbackPushforwardAdjunction C f)) F) (op V_1) ≫
      (obj (pullback C f) F).map (homOfLE hV_1).op ≫
        (obj (pullback C f) F).germ V x hV ≫ 𝟙 ((obj (pullback C f) F).stalk x)
⊢ NatTrans.app (NatTrans.app (Adjunction.unit (pullbackPushforwardAdjunction C f)) F) (op V_1) ≫
      (obj (pullback C f) F).map (homOfLE hV_1).op ≫
        (obj (pullback C f) F).germ V x hV ≫ stalkPullbackInv C f F x ≫ stalkPullbackHom C f F x =
    NatTrans.app (NatTrans.app (Adjunction.unit (pullbackPushforwardAdjunction C f)) F) (op V_1) ≫
      (obj (pullback C f) F).map (homOfLE hV_1).op ≫
        (obj (pullback C f) F).germ V x hV ≫ 𝟙 ((obj (pullback C f) F).stalk x)

Tactic:
dsimp
-- Module: Mathlib.Topology.Sheaves.Stalks
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : HasColimits.{v, u} C
X Y Z : TopCat
FC : C → C → Type u_1
CC : C → Type u_2
inst_2 : (X Y : C) → FunLike.{u_1 + 1, u_2 + 1, u_2 + 1} (FC X Y) (CC X) (CC Y)
f : X ⟶ Y
F : Presheaf C Y
x : (↑X : Type v)
V : Opens (↑X : Type v)
hV : x ∈ V
V_1 : Opens (↑Y : Type v)
hV_1 : unop (op V) ≤ (Opens.map f).obj V_1
ih :
  NatTrans.app (NatTrans.app (Adjunction.unit (pullbackPushforwardAdjunction C f)) F) (op V_1) ≫
      (obj (pullback C f) F).map (homOfLE hV_1).op ≫ germToPullbackStalk C f F V x hV ≫ stalkPullbackHom C f F x =
    NatTrans.app (NatTrans.app (Adjunction.unit (pullbackPushforwardAdjunction C f)) F) (op V_1) ≫
      (obj (pullback C f) F).map (homOfLE hV_1).op ≫
        (obj (pullback C f) F).germ V x hV ≫ 𝟙 ((obj (pullback C f) F).stalk x)
ih₁ :
  NatTrans.app (NatTrans.app (Adjunction.unit (pullbackPushforwardAdjunction C f)) F) (op V_1) ≫
      (obj (pullback C f) F).map (homOfLE hV_1).op ≫ germToPullbackStalk C f F V x hV ≫ stalkPullbackHom C f F x =
    NatTrans.app (NatTrans.app (Adjunction.unit (pullbackPushforwardAdjunction C f)) F) (op V_1) ≫
      (obj (pullback C f) F).map (homOfLE hV_1).op ≫ (obj (pullback C f) F).germ V x hV
ih₂ ih₃ :
  Eq (α := F.obj (op V_1) ⟶ (obj (pullback C f) F).stalk x)
    (NatTrans.app (NatTrans.app (Adjunction.unit (pullbackPushforwardAdjunction C f)) F) (op V_1) ≫
      (obj (pullback C f) F).map (homOfLE hV_1).op ≫ (obj (pullback C f) F).germ V x hV)
    (NatTrans.app (NatTrans.app (Adjunction.unit (pullbackPushforwardAdjunction C f)) F) (op V_1) ≫
      (obj (pullback C f) F).map (homOfLE hV_1).op ≫ (obj (pullback C f) F).germ V x hV)
⊢ NatTrans.app (NatTrans.app (Adjunction.unit (pullbackPushforwardAdjunction C f)) F) (op V_1) ≫
      (obj (pullback C f) F).map (homOfLE hV_1).op ≫
        (obj (pullback C f) F).germ V x hV ≫ stalkPullbackInv C f F x ≫ stalkPullbackHom C f F x =
    NatTrans.app (NatTrans.app (Adjunction.unit (pullbackPushforwardAdjunction C f)) F) (op V_1) ≫
      (obj (pullback C f) F).map (homOfLE hV_1).op ≫
        (obj (pullback C f) F).germ V x hV ≫ 𝟙 ((obj (pullback C f) F).stalk x)

Tactic:
rw [germ_stalkPullbackInv_assoc, Category.comp_id, germToPullbackStalk_stalkPullbackHom]

Premises used: TopCat.Presheaf.germ_stalkPullbackInv_assoc, TopCat.Presheaf.germToPullbackStalk_stalkPullbackHom, CategoryTheory.Category.comp_id
-- Module: Mathlib.Topology.Sheaves.Stalks
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : HasColimits.{v, u} C
X Y Z : TopCat
FC : C → C → Type u_1
CC : C → Type u_2
inst_2 : (X Y : C) → FunLike.{u_1 + 1, u_2 + 1, u_2 + 1} (FC X Y) (CC X) (CC Y)
F : Presheaf C X
x y : (↑X : Type v)
h : x ⤳ y
refine_1 :
  (U : (OpenNhds y)ᵒᵖ) →
    (obj (obj (whiskeringLeft (OpenNhds y)ᵒᵖ (Opens (↑X : Type v))ᵒᵖ C) (OpenNhds.inclusion y).op) F).obj U ⟶
      obj (obj (Functor.const (OpenNhds y)ᵒᵖ) ((stalkFunctor C x : Presheaf C X ⥤ C).1 F)) U
refine_2 :
  ∀ ⦃X_1 Y : (OpenNhds y)ᵒᵖ⦄ (f : X_1 ⟶ Y),
    Eq (α :=
      (obj (obj (whiskeringLeft (OpenNhds y)ᵒᵖ (Opens (↑X : Type v))ᵒᵖ C) (OpenNhds.inclusion y).op) F).obj X_1 ⟶
        obj (obj (Functor.const (OpenNhds y)ᵒᵖ) ((stalkFunctor C x : Presheaf C X ⥤ C).1 F)) Y)
      ((obj (obj (whiskeringLeft (OpenNhds y)ᵒᵖ (Opens (↑X : Type v))ᵒᵖ C) (OpenNhds.inclusion y).op) F).map f ≫
        (fun (U : (OpenNhds y)ᵒᵖ) ↦
            (colimit.ι ((OpenNhds.inclusion x).op ⋙ F)
                (op (Subtype.mk (p := fun (U : Opens (↑X : Type v)) ↦ x ∈ U) (↑(unop U) : Opens (↑X : Type v)) ⋯)) :
              ((OpenNhds.inclusion x).op ⋙ F).obj
                  (op (Subtype.mk (p := fun (U : Opens (↑X : Type v)) ↦ x ∈ U) (↑(unop U) : Opens (↑X : Type v)) ⋯)) ⟶
                colimit ((OpenNhds.inclusion x).op ⋙ F)))
          Y)
      ((fun (U : (OpenNhds y)ᵒᵖ) ↦
            (colimit.ι ((OpenNhds.inclusion x).op ⋙ F)
                (op (Subtype.mk (p := fun (U : Opens (↑X : Type v)) ↦ x ∈ U) (↑(unop U) : Opens (↑X : Type v)) ⋯)) :
              ((OpenNhds.inclusion x).op ⋙ F).obj
                  (op (Subtype.mk (p := fun (U : Opens (↑X : Type v)) ↦ x ∈ U) (↑(unop U) : Opens (↑X : Type v)) ⋯)) ⟶
                colimit ((OpenNhds.inclusion x).op ⋙ F)))
          X_1 ≫
        map (obj (Functor.const (OpenNhds y)ᵒᵖ) ((stalkFunctor C x : Presheaf C X ⥤ C).1 F)) f)
⊢ F.stalk y ⟶ F.stalk x

Tactic:
refine colimit.desc _ ⟨_, fun U => ?_, ?_⟩

Premises used: CategoryTheory.Limits.Cocone.mk, CategoryTheory.NatTrans.mk, CategoryTheory.Limits.colimit.desc
-- Module: Mathlib.Topology.Sheaves.Stalks
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : HasColimits.{v, u} C
X Y Z : TopCat
FC : C → C → Type u_1
CC : C → Type u_2
inst_2 : (X Y : C) → FunLike.{u_1 + 1, u_2 + 1, u_2 + 1} (FC X Y) (CC X) (CC Y)
F : Presheaf C X
x y : (↑X : Type v)
h : x ⤳ y
U : (OpenNhds y)ᵒᵖ
⊢ (obj (obj (whiskeringLeft (OpenNhds y)ᵒᵖ (Opens (↑X : Type v))ᵒᵖ C) (OpenNhds.inclusion y).op) F).obj U ⟶
    obj (obj (Functor.const (OpenNhds y)ᵒᵖ) ((stalkFunctor C x : Presheaf C X ⥤ C).1 F)) U

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 :)⟩)

Premises used: CategoryTheory.Functor.op, CategoryTheory.Functor.comp, Subtype.mk, Opposite.op, Iff.mp, TopologicalSpace.Opens.is_open', specializes_iff_forall_open, Subtype.val, Opposite.unop, TopologicalSpace.OpenNhds.inclusion, Subtype.property, CategoryTheory.Limits.colimit.ι
-- Module: Mathlib.Topology.Sheaves.Stalks
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : HasColimits.{v, u} C
X Y Z : TopCat
FC : C → C → Type u_1
CC : C → Type u_2
inst_2 : (X Y : C) → FunLike.{u_1 + 1, u_2 + 1, u_2 + 1} (FC X Y) (CC X) (CC Y)
F : Presheaf C X
x y : (↑X : Type v)
h : x ⤳ y
U V : (OpenNhds y)ᵒᵖ
i : U ⟶ V
refine_2 :
  Eq (α := F.obj (op ((OpenNhds.inclusion y).obj (unop U))) ⟶ F.stalk x)
    (F.map ((OpenNhds.inclusion y).map i.unop).op ≫
      colimit.ι ((OpenNhds.inclusion x).op ⋙ F)
        (op (Subtype.mk (p := fun (U : Opens (↑X : Type v)) ↦ x ∈ U) (↑(unop V) : Opens (↑X : Type v)) ⋯)))
    (colimit.ι ((OpenNhds.inclusion x).op ⋙ F)
        (op (Subtype.mk (p := fun (U : Opens (↑X : Type v)) ↦ x ∈ U) (↑(unop U) : Opens (↑X : Type v)) ⋯)) ≫
      𝟙 (F.stalk x))
⊢ Eq (α :=
    (obj (obj (whiskeringLeft (OpenNhds y)ᵒᵖ (Opens (↑X : Type v))ᵒᵖ C) (OpenNhds.inclusion y).op) F).obj U ⟶
      obj (obj (Functor.const (OpenNhds y)ᵒᵖ) ((stalkFunctor C x : Presheaf C X ⥤ C).1 F)) V)
    ((obj (obj (whiskeringLeft (OpenNhds y)ᵒᵖ (Opens (↑X : Type v))ᵒᵖ C) (OpenNhds.inclusion y).op) F).map i ≫
      (fun (U : (OpenNhds y)ᵒᵖ) ↦
          (colimit.ι ((OpenNhds.inclusion x).op ⋙ F)
              (op (Subtype.mk (p := fun (U : Opens (↑X : Type v)) ↦ x ∈ U) (↑(unop U) : Opens (↑X : Type v)) ⋯)) :
            ((OpenNhds.inclusion x).op ⋙ F).obj
                (op (Subtype.mk (p := fun (U : Opens (↑X : Type v)) ↦ x ∈ U) (↑(unop U) : Opens (↑X : Type v)) ⋯)) ⟶
              colimit ((OpenNhds.inclusion x).op ⋙ F)))
        V)
    ((fun (U : (OpenNhds y)ᵒᵖ) ↦
          (colimit.ι ((OpenNhds.inclusion x).op ⋙ F)
              (op (Subtype.mk (p := fun (U : Opens (↑X : Type v)) ↦ x ∈ U) (↑(unop U) : Opens (↑X : Type v)) ⋯)) :
            ((OpenNhds.inclusion x).op ⋙ F).obj
                (op (Subtype.mk (p := fun (U : Opens (↑X : Type v)) ↦ x ∈ U) (↑(unop U) : Opens (↑X : Type v)) ⋯)) ⟶
              colimit ((OpenNhds.inclusion x).op ⋙ F)))
        U ≫
      map (obj (Functor.const (OpenNhds y)ᵒᵖ) ((stalkFunctor C x : Presheaf C X ⥤ C).1 F)) i)

Tactic:
dsimp
-- Module: Mathlib.Topology.Sheaves.Stalks
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : HasColimits.{v, u} C
X Y Z : TopCat
FC : C → C → Type u_1
CC : C → Type u_2
inst_2 : (X Y : C) → FunLike.{u_1 + 1, u_2 + 1, u_2 + 1} (FC X Y) (CC X) (CC Y)
F : Presheaf C X
x y : (↑X : Type v)
h : x ⤳ y
U V : (OpenNhds y)ᵒᵖ
i : U ⟶ V
refine_2 :
  Eq (α := F.obj (op ((OpenNhds.inclusion y).obj (unop U))) ⟶ F.stalk x)
    (F.map ((OpenNhds.inclusion y).map i.unop).op ≫
      colimit.ι ((OpenNhds.inclusion x).op ⋙ F)
        (op (Subtype.mk (p := fun (U : Opens (↑X : Type v)) ↦ x ∈ U) (↑(unop V) : Opens (↑X : Type v)) ⋯)))
    (colimit.ι ((OpenNhds.inclusion x).op ⋙ F)
      (op (Subtype.mk (p := fun (U : Opens (↑X : Type v)) ↦ x ∈ U) (↑(unop U) : Opens (↑X : Type v)) ⋯)))
⊢ Eq (α := F.obj (op ((OpenNhds.inclusion y).obj (unop U))) ⟶ F.stalk x)
    (F.map ((OpenNhds.inclusion y).map i.unop).op ≫
      colimit.ι ((OpenNhds.inclusion x).op ⋙ F)
        (op (Subtype.mk (p := fun (U : Opens (↑X : Type v)) ↦ x ∈ U) (↑(unop V) : Opens (↑X : Type v)) ⋯)))
    (colimit.ι ((OpenNhds.inclusion x).op ⋙ F)
        (op (Subtype.mk (p := fun (U : Opens (↑X : Type v)) ↦ x ∈ U) (↑(unop U) : Opens (↑X : Type v)) ⋯)) ≫
      𝟙 (F.stalk x))

Tactic:
rw [Category.comp_id]

Premises used: CategoryTheory.Category.comp_id
-- Module: Mathlib.Topology.Sheaves.Stalks
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : HasColimits.{v, u} C
X Y Z : TopCat
FC : C → C → Type u_1
CC : C → Type u_2
inst_2 : (X Y : C) → FunLike.{u_1 + 1, u_2 + 1, u_2 + 1} (FC X Y) (CC X) (CC Y)
F : Presheaf C X
x y : (↑X : Type v)
h : x ⤳ y
U V : (OpenNhds y)ᵒᵖ
i : U ⟶ V
U' : OpenNhds x := Subtype.mk (p := fun (U : Opens (↑X : Type v)) ↦ x ∈ U) (↑(unop U) : Opens (↑X : Type v)) ⋯
V' : OpenNhds x := Subtype.mk (p := fun (U : Opens (↑X : Type v)) ↦ x ∈ U) (↑(unop V) : Opens (↑X : Type v)) ⋯
⊢ Eq (α := F.obj (op ((OpenNhds.inclusion y).obj (unop U))) ⟶ F.stalk x)
    (F.map ((OpenNhds.inclusion y).map i.unop).op ≫
      colimit.ι ((OpenNhds.inclusion x).op ⋙ F)
        (op (Subtype.mk (p := fun (U : Opens (↑X : Type v)) ↦ x ∈ U) (↑(unop V) : Opens (↑X : Type v)) ⋯)))
    (colimit.ι ((OpenNhds.inclusion x).op ⋙ F)
      (op (Subtype.mk (p := fun (U : Opens (↑X : Type v)) ↦ x ∈ U) (↑(unop U) : Opens (↑X : Type v)) ⋯)))

Tactic:
colimit.w ((OpenNhds.inclusion x).op ⋙ F) (show V' ⟶ U' from i.unop).op

Premises used: CategoryTheory.Functor.op, Set.Mem, Quiver.Hom.op, CategoryTheory.Functor.comp, Quiver.Hom.unop, TopCat.Presheaf.stalk, CategoryTheory.Limits.colimit, CategoryTheory.Limits.colimit.w, TopologicalSpace.OpenNhds.inclusion
-- Module: Mathlib.Topology.Sheaves.Stalks
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : HasColimits.{v, u} C
X Y Z : TopCat
FC : C → C → Type u_1
CC : C → Type u_2
inst_2 : (X Y : C) → FunLike.{u_1 + 1, u_2 + 1, u_2 + 1} (FC X Y) (CC X) (CC Y)
F : Presheaf C X
x y : (↑X : Type v)
h : x ⤳ y
U V : (OpenNhds y)ᵒᵖ
i : U ⟶ V
U' : OpenNhds x := Subtype.mk (p := fun (U : Opens (↑X : Type v)) ↦ x ∈ U) (↑(unop U) : Opens (↑X : Type v)) ⋯
V' : OpenNhds x := Subtype.mk (p := fun (U : Opens (↑X : Type v)) ↦ x ∈ U) (↑(unop V) : Opens (↑X : Type v)) ⋯
⊢ Eq (α := F.obj (op ((OpenNhds.inclusion y).obj (unop U))) ⟶ F.stalk x)
    (F.map ((OpenNhds.inclusion y).map i.unop).op ≫
      colimit.ι ((OpenNhds.inclusion x).op ⋙ F)
        (op (Subtype.mk (p := fun (U : Opens (↑X : Type v)) ↦ x ∈ U) (↑(unop V) : Opens (↑X : Type v)) ⋯)))
    (colimit.ι ((OpenNhds.inclusion x).op ⋙ F)
      (op (Subtype.mk (p := fun (U : Opens (↑X : Type v)) ↦ x ∈ U) (↑(unop U) : Opens (↑X : Type v)) ⋯)))

Tactic:
exact colimit.w ((OpenNhds.inclusion x).op ⋙ F) (show V' ⟶ U' from i.unop).op

Premises used: CategoryTheory.Functor.op, Quiver.Hom.op, CategoryTheory.Functor.comp, Quiver.Hom.unop, CategoryTheory.Limits.colimit.w, TopologicalSpace.OpenNhds.inclusion
-- Module: Mathlib.Topology.Sheaves.Stalks
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : HasColimits.{v, u} C
X : TopCat
F : Presheaf C X
U : Opens (↑X : Type v)
y : (↑X : Type v)
hy : y ∈ U
x : (↑X : Type v)
h : x ⤳ y
⊢ Eq (α := F.obj (op U) ⟶ F.stalk x) (F.germ U y hy ≫ F.stalkSpecializes h) (F.germ U x ⋯)

Tactic:
colimit.ι_desc _ _

Premises used: TopCat.Presheaf.stalkSpecializes, CategoryTheory.Functor.comp, TopCat.Presheaf.stalk, CategoryTheory.Limits.colimit, TopCat.Presheaf.germ, CategoryTheory.Limits.colimit.ι_desc, CategoryTheory.Limits.colimit.ι
-- Module: Mathlib.Topology.Sheaves.Stalks
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : HasColimits.{v, u} C
X : TopCat
F : Presheaf C X
x : (↑X : Type v)
ih : ∀ (U : Opens (↑X : Type v)) (hxU : x ∈ U), F.germ U x hxU ≫ F.stalkSpecializes ⋯ = F.germ U x hxU ≫ 𝟙 (F.stalk x)
⊢ F.stalkSpecializes ⋯ = 𝟙 (F.stalk x)

Tactic:
ext
-- Module: Mathlib.Topology.Sheaves.Stalks
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : HasColimits.{v, u} C
X : TopCat
F : Presheaf C X
x : (↑X : Type v)
U : Opens (↑X : Type v)
hxU : x ∈ U
⊢ F.germ U x hxU ≫ F.stalkSpecializes ⋯ = F.germ U x hxU ≫ 𝟙 (F.stalk x)

Tactic:
simp
-- Module: Mathlib.Topology.Sheaves.Stalks
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : HasColimits.{v, u} C
X : TopCat
F : Presheaf C X
x y z : (↑X : Type v)
h : x ⤳ y
h' : y ⤳ z
ih :
  ∀ (U : Opens (↑X : Type v)) (hxU : z ∈ U),
    Eq (α := F.obj (op U) ⟶ F.stalk x) (F.germ U z hxU ≫ F.stalkSpecializes h' ≫ F.stalkSpecializes h)
      (F.germ U z hxU ≫ F.stalkSpecializes ⋯)
⊢ Eq (α := F.stalk z ⟶ F.stalk x) (F.stalkSpecializes h' ≫ F.stalkSpecializes h) (F.stalkSpecializes ⋯)

Tactic:
ext
-- Module: Mathlib.Topology.Sheaves.Stalks
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : HasColimits.{v, u} C
X : TopCat
F : Presheaf C X
x y z : (↑X : Type v)
h : x ⤳ y
h' : y ⤳ z
U : Opens (↑X : Type v)
hxU : z ∈ U
⊢ Eq (α := F.obj (op U) ⟶ F.stalk x) (F.germ U z hxU ≫ F.stalkSpecializes h' ≫ F.stalkSpecializes h)
    (F.germ U z hxU ≫ F.stalkSpecializes ⋯)

Tactic:
simp
-- Module: Mathlib.Topology.Sheaves.Stalks
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : HasColimits.{v, u} C
X : TopCat
F G : Presheaf C X
f : F ⟶ G
x y : (↑X : Type v)
h : x ⤳ y
ih :
  ∀ (U : Opens (↑X : Type v)) (hxU : y ∈ U),
    Eq (α := F.obj (op U) ⟶ obj (stalkFunctor C x) G) (F.germ U y hxU ≫ F.stalkSpecializes h ≫ map (stalkFunctor C x) f)
      (F.germ U y hxU ≫ map (stalkFunctor C y) f ≫ G.stalkSpecializes h)
⊢ Eq (α := F.stalk y ⟶ obj (stalkFunctor C x) G) (F.stalkSpecializes h ≫ map (stalkFunctor C x) f)
    (map (stalkFunctor C y) f ≫ G.stalkSpecializes h)

Tactic:
ext
-- Module: Mathlib.Topology.Sheaves.Stalks
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : HasColimits.{v, u} C
X : TopCat
F G : Presheaf C X
f : F ⟶ G
x y : (↑X : Type v)
h : x ⤳ y
U : Opens (↑X : Type v)
hxU : y ∈ U
⊢ Eq (α := F.obj (op U) ⟶ obj (stalkFunctor C x) G) (F.germ U y hxU ≫ F.stalkSpecializes h ≫ map (stalkFunctor C x) f)
    (F.germ U y hxU ≫ map (stalkFunctor C y) f ≫ G.stalkSpecializes h)

Tactic:
simp
-- Module: Mathlib.Topology.Sheaves.Stalks
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : HasColimits.{v, u} C
X Y : TopCat
f : X ⟶ Y
F : Presheaf C X
x y : (↑X : Type v)
h : x ⤳ y
ih :
  ∀ (U : Opens (↑Y : Type v))
    (hxU : Membership.mem (α := (↑Y : Type v)) U ((Hom.hom f : (a : (↑X : Type v)) → (↑Y : Type v)) y : (↑Y : Type v))),
    Eq (α := (obj (pushforward C f) F).obj (op U) ⟶ F.stalk x)
      ((obj (pushforward C f) F).germ U ((Hom.hom f : (a : (↑X : Type v)) → (↑Y : Type v)) y) hxU ≫
        (obj (pushforward C f) F).stalkSpecializes ⋯ ≫ stalkPushforward C f F x)
      ((obj (pushforward C f) F).germ U ((Hom.hom f : (a : (↑X : Type v)) → (↑Y : Type v)) y) hxU ≫
        stalkPushforward C f F y ≫ F.stalkSpecializes h)
⊢ Eq (α := (obj (pushforward C f) F).stalk ((Hom.hom f : (a : (↑X : Type v)) → (↑Y : Type v)) y) ⟶ F.stalk x)
    ((obj (pushforward C f) F).stalkSpecializes ⋯ ≫ stalkPushforward C f F x)
    (stalkPushforward C f F y ≫ F.stalkSpecializes h)

Tactic:
ext
-- Module: Mathlib.Topology.Sheaves.Stalks
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : HasColimits.{v, u} C
X Y : TopCat
f : X ⟶ Y
F : Presheaf C X
x y : (↑X : Type v)
h : x ⤳ y
U : Opens (↑Y : Type v)
hxU : Membership.mem (α := (↑Y : Type v)) U ((Hom.hom f : (a : (↑X : Type v)) → (↑Y : Type v)) y : (↑Y : Type v))
⊢ Eq (α := (obj (pushforward C f) F).obj (op U) ⟶ F.stalk x)
    ((obj (pushforward C f) F).germ U ((Hom.hom f : (a : (↑X : Type v)) → (↑Y : Type v)) y) hxU ≫
      (obj (pushforward C f) F).stalkSpecializes ⋯ ≫ stalkPushforward C f F x)
    ((obj (pushforward C f) F).germ U ((Hom.hom f : (a : (↑X : Type v)) → (↑Y : Type v)) y) hxU ≫
      stalkPushforward C f F y ≫ F.stalkSpecializes h)

Tactic:
simp
-- Module: Mathlib.Topology.Sheaves.Stalks
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : HasColimits.{v, u} C
X Y Z : TopCat
FC : C → C → Type u_1
CC : C → Type u_2
inst_2 : (X Y : C) → FunLike.{u_1 + 1, u_2 + 1, u_2 + 1} (FC X Y) (CC X) (CC Y)
F : Presheaf C X
x y : (↑X : Type v)
e : Inseparable x y
⊢ F.stalkSpecializes ⋯ ≫ F.stalkSpecializes ⋯ = 𝟙 (F.stalk x)

Tactic:
simp
-- Module: Mathlib.Topology.Sheaves.Stalks
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : HasColimits.{v, u} C
X Y Z : TopCat
FC : C → C → Type u_1
CC : C → Type u_2
inst_2 : (X Y : C) → FunLike.{u_1 + 1, u_2 + 1, u_2 + 1} (FC X Y) (CC X) (CC Y)
F : Presheaf C X
x y : (↑X : Type v)
e : Inseparable x y
⊢ F.stalkSpecializes ⋯ ≫ F.stalkSpecializes ⋯ = 𝟙 (F.stalk y)

Tactic:
simp
-- 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))
F : Presheaf C X
x : (↑X : Type v)
t : ToType (FC := FC) (CC := CC) (F.stalk x)
obtain :
  ∀ (U : (OpenNhds x)ᵒᵖ)
    (s :
      (obj (obj (whiskeringLeft (OpenNhds x)ᵒᵖ (Opens (↑X : Type v))ᵒᵖ C) (OpenNhds.inclusion x).op) F ⋙
            forget C (FC := FC) (CC := CC)).obj
        U)
    (e :
      (ConcreteCategory.hom (FC := TypeCat.Fun)
              (NatTrans.app
                (Cocone.ι
                  (mapCocone (forget C (FC := FC) (CC := CC))
                    (colimit.cocone
                      (obj (obj (whiskeringLeft (OpenNhds x)ᵒᵖ (Opens (↑X : Type v))ᵒᵖ C) (OpenNhds.inclusion x).op)
                        F))))
                U) :
            (a :
                (fun (X : Type v) ↦ X)
                  ((((whiskeringLeft (OpenNhds x)ᵒᵖ (Opens (↑X : Type v))ᵒᵖ C).obj (OpenNhds.inclusion x).op).obj F ⋙
                        forget C).obj
                    U)) →
              (fun (X : Type v) ↦ X)
                (((Functor.const (OpenNhds x)ᵒᵖ).obj
                      ((forget C).mapCocone
                          (colimit.cocone
                            (((whiskeringLeft (OpenNhds x)ᵒᵖ (Opens (↑X : Type v))ᵒᵖ C).obj
                                  (OpenNhds.inclusion x).op).obj
                              F))).pt).obj
                  U))
          s =
        t),
    ∃ (U : Opens (↑X : Type v)) (m : x ∈ U) (s : ToType (FC := FC) (CC := CC) (F.obj (op U))),
      (ConcreteCategory.hom (FC := FC) (CC := CC) (F.germ U x m) : (a : CC (F.obj (op U))) → CC (F.stalk x)) s = t
⊢ ∃ (U : Opens (↑X : Type v)) (m : x ∈ U) (s : ToType (FC := FC) (CC := CC) (F.obj (op U))),
    (ConcreteCategory.hom (FC := FC) (CC := CC) (F.germ U x m) : (a : CC (F.obj (op U))) → CC (F.stalk x)) s = t

Tactic:
obtain ⟨U, s, e⟩ :=
    Types.jointly_surjective.{v, v} _ (isColimitOfPreserves (forget C) (colimit.isColimit _)) t