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 : TopCat
f : X ⟶ Y
F : Presheaf C Y
x : (↑X : Type v)
U : Opens (↑Y : Type v)
hU :
  Membership.mem (α := (↑Y : Type v)) U
    ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := carrier) f :
          (a : (↑X : Type v)) → (↑Y : Type v))
        x :
      (↑Y : Type v))
⊢ F.germ U
        ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := carrier) f :
            (a : (↑X : Type v)) → (↑Y : Type v))
          x)
        hU ≫
      stalkPullbackHom C f F x =
    NatTrans.app (NatTrans.app (Adjunction.unit (pullbackPushforwardAdjunction C f)) F) (op U) ≫
      (obj (pullback C f) F).germ ((Opens.map f).obj U) x hU

Tactic:
simp [stalkPullbackHom, germ, stalkFunctor, stalkPushforward]

Premises used: TopCat.Presheaf.germ, TopCat.Presheaf.stalkFunctor, TopCat.Presheaf.stalkPullbackHom, TopCat.Presheaf.stalkPushforward
-- 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
U : Opens (↑X : Type v)
x : (↑X : Type v)
hx : x ∈ U
x_1 x_2 : CostructuredArrow (Opens.map f).op (op U)
i : x_1 ⟶ x_2
⊢ Eq (α :=
    obj (CostructuredArrow.proj (Opens.map f).op (op U) ⋙ F) x_1 ⟶
      obj
        (obj (Functor.const (CostructuredArrow (Opens.map f).op (op U)))
          (F.stalk
            ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := carrier) f :
                (a : (↑X : Type v)) → (↑Y : Type v))
              x)))
        x_2)
    (map (CostructuredArrow.proj (Opens.map f).op (op U) ⋙ F) i ≫
      F.germ (unop x_2.1)
        ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := carrier) f :
            (a : (↑X : Type v)) → (↑Y : Type v))
          x)
        ⋯)
    (CategoryStruct.comp (Y :=
      obj
        (obj (Functor.const (CostructuredArrow (Opens.map f).op (op U)))
          (F.stalk
            ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := carrier) f :
                (a : (↑X : Type v)) → (↑Y : Type v))
              x)))
        x_1)
      (F.germ (unop x_1.1)
          ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := carrier) f :
              (a : (↑X : Type v)) → (↑Y : Type v))
            x)
          ⋯ :
        F.obj (op (unop x_1.1)) ⟶
          F.stalk
            ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := carrier) f :
                (a : (↑X : Type v)) → (↑Y : Type v))
              x))
      (map
        (obj (Functor.const (CostructuredArrow (Opens.map f).op (op U)))
          (F.stalk
            ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := carrier) f :
                (a : (↑X : Type v)) → (↑Y : Type v))
              x)))
        i))

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
Z : C
f : X ⟶ Y
F : Presheaf C Y
U : (Opens (↑X : Type v))ᵒᵖ
φ ψ : (obj (pullback C f) F).obj U ⟶ Z
h :
  ∀ (V : Opens (↑Y : Type v)) (hV : unop U ≤ (Opens.map f).obj V),
    NatTrans.app (NatTrans.app (Adjunction.unit (pullbackPushforwardAdjunction C f)) F) (op V) ≫
        (obj (pullback C f) F).map (homOfLE hV).op ≫ φ =
      NatTrans.app (NatTrans.app (Adjunction.unit (pullbackPushforwardAdjunction C f)) F) (op V) ≫
        (obj (pullback C f) F).map (homOfLE hV).op ≫ ψ
apply :
  ∀ (j : CostructuredArrow (Opens.map f).op U),
    ((LeftExtension.mk ((Opens.map f).op.leftKanExtension F) ((Opens.map f).op.leftKanExtensionUnit F)).coconeAt
                U).ι.app
          j ≫
        φ =
      ((LeftExtension.mk ((Opens.map f).op.leftKanExtension F) ((Opens.map f).op.leftKanExtensionUnit F)).coconeAt
                U).ι.app
          j ≫
        ψ
⊢ φ = ψ

Tactic:
apply ((Opens.map f).op.isPointwiseLeftKanExtensionLeftKanExtensionUnit F _).hom_ext

Premises used: CategoryTheory.Functor.op, CategoryTheory.Limits.IsColimit.hom_ext, TopologicalSpace.Opens.map, CategoryTheory.Functor.isPointwiseLeftKanExtensionLeftKanExtensionUnit
-- Module: Mathlib.Topology.Sheaves.Stalks
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : HasColimits.{v, u} C
X Y : TopCat
Z : C
f : X ⟶ Y
F : Presheaf C Y
U : (Opens (↑X : Type v))ᵒᵖ
φ ψ : (obj (pullback C f) F).obj U ⟶ Z
h :
  ∀ (V : Opens (↑Y : Type v)) (hV : unop U ≤ (Opens.map f).obj V),
    NatTrans.app (NatTrans.app (Adjunction.unit (pullbackPushforwardAdjunction C f)) F) (op V) ≫
        (obj (pullback C f) F).map (homOfLE hV).op ≫ φ =
      NatTrans.app (NatTrans.app (Adjunction.unit (pullbackPushforwardAdjunction C f)) F) (op V) ≫
        (obj (pullback C f) F).map (homOfLE hV).op ≫ ψ
V : Opens (↑Y : Type v)
as : PUnit.{1}
b : unop (obj (fromPUnit U) { as := as : Discrete PUnit.{1} }) ⟶ unop ((Opens.map f).op.obj (op V))
⊢ Eq (α :=
    (CostructuredArrow.proj (Opens.map f).op U ⋙ F).obj
        { left := op V, right := { as := as : Discrete PUnit.{1} }, hom := op b :
          Comma (Opens.map f).op (fromPUnit U) } ⟶
      Z)
    (((LeftExtension.mk ((Opens.map f).op.leftKanExtension F) ((Opens.map f).op.leftKanExtensionUnit F)).coconeAt
              U).ι.app
        { left := op V, right := { as := as }, hom := op b } ≫
      φ)
    (((LeftExtension.mk ((Opens.map f).op.leftKanExtension F) ((Opens.map f).op.leftKanExtensionUnit F)).coconeAt
              U).ι.app
        { left := op V, right := { as := as }, hom := op b } ≫
      ψ)

Tactic:
simpa [pullbackPushforwardAdjunction, Functor.lanAdjunction_unit]
    using h V (leOfHom b)

Premises used: TopCat.Presheaf.pullbackPushforwardAdjunction, CategoryTheory.Functor.lanAdjunction_unit, CategoryTheory.leOfHom
-- 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 Y
U : Opens (↑X : Type v)
x : (↑X : Type v)
hx : x ∈ U
V : Opens (↑Y : Type v)
hV : U ≤ (Opens.map f).obj V
⊢ NatTrans.app (NatTrans.app (Adjunction.unit (pullbackPushforwardAdjunction C f)) F) (op V) ≫
      (obj (pullback C f) F).map (homOfLE hV).op ≫ germToPullbackStalk C f F U x hx =
    F.germ V
      ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := carrier) f :
          (a : (↑X : Type v)) → (↑Y : Type v))
        x)
      ⋯

Tactic:
simpa [pullbackPushforwardAdjunction] using
    ((Opens.map f).op.isPointwiseLeftKanExtensionLeftKanExtensionUnit F (op U)).fac _
      (CostructuredArrow.mk (homOfLE hV).op)

Premises used: CategoryTheory.Functor.op, CategoryTheory.Limits.IsColimit.fac, Quiver.Hom.op, TopCat.Presheaf.pullbackPushforwardAdjunction, Opposite.op, CategoryTheory.homOfLE, CategoryTheory.CostructuredArrow.mk, TopologicalSpace.Opens.map, CategoryTheory.Functor.isPointwiseLeftKanExtensionLeftKanExtensionUnit
-- 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 Y
U : Opens (↑X : Type v)
x : (↑X : Type v)
hx : x ∈ U
h :
  ∀ (V : Opens (↑Y : Type v)) (hV : unop (op U) ≤ (Opens.map f).obj V),
    NatTrans.app (NatTrans.app (Adjunction.unit (pullbackPushforwardAdjunction C f)) F) (op V) ≫
        (obj (pullback C f) F).map (homOfLE hV).op ≫ germToPullbackStalk C f F U x hx ≫ stalkPullbackHom C f F x =
      NatTrans.app (NatTrans.app (Adjunction.unit (pullbackPushforwardAdjunction C f)) F) (op V) ≫
        (obj (pullback C f) F).map (homOfLE hV).op ≫ (obj (pullback C f) F).germ U x hx
⊢ germToPullbackStalk C f F U x hx ≫ stalkPullbackHom C f F x = (obj (pullback C f) F).germ U x hx

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 : TopCat
f : X ⟶ Y
F : Presheaf C Y
U : Opens (↑X : Type v)
x : (↑X : Type v)
hx : x ∈ U
V : Opens (↑Y : Type v)
hV : unop (op U) ≤ (Opens.map f).obj V
h :
  NatTrans.app (NatTrans.app (Adjunction.unit (pullbackPushforwardAdjunction C f)) F) (op V) ≫
      (obj (pullback C f) F).map (homOfLE hV).op ≫ germToPullbackStalk C f F U x hx ≫ stalkPullbackHom C f F x =
    NatTrans.app (NatTrans.app (Adjunction.unit (pullbackPushforwardAdjunction C f)) F) (op V) ≫
      (obj (pullback C f) F).map (homOfLE hV).op ≫ (obj (pullback C f) F).germ U x hx
⊢ NatTrans.app (NatTrans.app (Adjunction.unit (pullbackPushforwardAdjunction C f)) F) (op V) ≫
      (obj (pullback C f) F).map (homOfLE hV).op ≫ germToPullbackStalk C f F U x hx ≫ stalkPullbackHom C f F x =
    NatTrans.app (NatTrans.app (Adjunction.unit (pullbackPushforwardAdjunction C f)) F) (op V) ≫
      (obj (pullback C f) F).map (homOfLE hV).op ≫ (obj (pullback C f) F).germ U x hx

Tactic:
dsimp
-- 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 Y
U : Opens (↑X : Type v)
x : (↑X : Type v)
hx : x ∈ U
V : Opens (↑Y : Type v)
hV : unop (op U) ≤ (Opens.map f).obj V
⊢ NatTrans.app (NatTrans.app (Adjunction.unit (pullbackPushforwardAdjunction C f)) F) (op V) ≫
      (obj (pullback C f) F).map (homOfLE hV).op ≫ germToPullbackStalk C f F U x hx ≫ stalkPullbackHom C f F x =
    NatTrans.app (NatTrans.app (Adjunction.unit (pullbackPushforwardAdjunction C f)) F) (op V) ≫
      (obj (pullback C f) F).map (homOfLE hV).op ≫ (obj (pullback C f) F).germ U x hx

Tactic:
simp only [pullbackPushforwardAdjunction_unit_pullback_map_germToPullbackStalk_assoc,
    germ_stalkPullbackHom, germ_res]

Premises used: TopCat.Presheaf.germ_stalkPullbackHom, TopCat.Presheaf.germ_res, TopCat.Presheaf.pullbackPushforwardAdjunction_unit_pullback_map_germToPullbackStalk_assoc
-- 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 Y
V : (Opens (↑Y : Type v))ᵒᵖ
x : (↑X : Type v)
hx :
  Membership.mem (α := (↑Y : Type v)) (unop V)
    ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := carrier) f :
          (a : (↑X : Type v)) → (↑Y : Type v))
        x :
      (↑Y : Type v))
⊢ (Opens.map f).obj (unop V) ≤ (Opens.map f).obj (unop V)

Tactic:
rfl
-- 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 Y
V : (Opens (↑Y : Type v))ᵒᵖ
x : (↑X : Type v)
hx :
  Membership.mem (α := (↑Y : Type v)) (unop V)
    ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := carrier) f :
          (a : (↑X : Type v)) → (↑Y : Type v))
        x :
      (↑Y : Type v))
⊢ Eq (α :=
    (obj (𝟭 (Presheaf C Y)) F).obj V ⟶
      F.stalk
        ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := carrier) f :
            (a : (↑X : Type v)) → (↑Y : Type v))
          x))
    (NatTrans.app (NatTrans.app (Adjunction.unit (pullbackPushforwardAdjunction C f)) F) V ≫
      germToPullbackStalk C f F ((Opens.map f).obj (unop V)) x hx)
    (F.germ (unop V)
      ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := carrier) f :
          (a : (↑X : Type v)) → (↑Y : Type v))
        x)
      hx)

Tactic:
simpa using pullbackPushforwardAdjunction_unit_pullback_map_germToPullbackStalk
    C f F ((Opens.map f).obj V.unop) x hx V.unop (by rfl)

Premises used: TopCat.Presheaf.pullbackPushforwardAdjunction_unit_pullback_map_germToPullbackStalk, TopologicalSpace.Opens.map, Opposite.unop, CategoryTheory.Functor.obj
-- 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)
U V : (OpenNhds x)ᵒᵖ
i : U ⟶ V
dsimp :
  (obj (pullback C f) F).map ((OpenNhds.inclusion x).map i.unop).op ≫
      germToPullbackStalk C f F (↑(unop V) : Opens (↑X : Type v)) x ⋯ =
    germToPullbackStalk C f F (↑(unop U) : Opens (↑X : Type v)) x ⋯ ≫
      𝟙
        (F.stalk
          ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := carrier) f :
              (a : (↑X : Type v)) → (↑Y : Type v))
            x))
⊢ ((OpenNhds.inclusion x).op ⋙ obj (pullback C f) F).map i ≫
      germToPullbackStalk C f F (↑(unop V) : Opens (↑X : Type v)) x ⋯ =
    germToPullbackStalk C f F (↑(unop U) : Opens (↑X : Type v)) x ⋯ ≫
      map
        (obj (Functor.const (OpenNhds x)ᵒᵖ)
          (F.stalk
            ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := carrier) f :
                (a : (↑X : Type v)) → (↑Y : Type v))
              x)))
        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 : X ⟶ Y
F : Presheaf C Y
x : (↑X : Type v)
U V : (OpenNhds x)ᵒᵖ
i : U ⟶ V
h :
  ∀ (W : Opens (↑Y : Type v)) (hW : unop (op ((OpenNhds.inclusion x).obj (unop U))) ≤ (Opens.map f).obj W),
    NatTrans.app (NatTrans.app (Adjunction.unit (pullbackPushforwardAdjunction C f)) F) (op W) ≫
        (obj (pullback C f) F).map (homOfLE hW).op ≫
          (obj (pullback C f) F).map ((OpenNhds.inclusion x).map i.unop).op ≫
            germToPullbackStalk C f F (↑(unop V) : Opens (↑X : Type v)) x ⋯ =
      NatTrans.app (NatTrans.app (Adjunction.unit (pullbackPushforwardAdjunction C f)) F) (op W) ≫
        (obj (pullback C f) F).map (homOfLE hW).op ≫
          germToPullbackStalk C f F (↑(unop U) : Opens (↑X : Type v)) x ⋯ ≫
            𝟙
              (F.stalk
                ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := carrier)
                      f :
                    (a : (↑X : Type v)) → (↑Y : Type v))
                  x))
⊢ (obj (pullback C f) F).map ((OpenNhds.inclusion x).map i.unop).op ≫
      germToPullbackStalk C f F (↑(unop V) : Opens (↑X : Type v)) x ⋯ =
    germToPullbackStalk C f F (↑(unop U) : Opens (↑X : Type v)) x ⋯ ≫
      𝟙
        (F.stalk
          ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := carrier) f :
              (a : (↑X : Type v)) → (↑Y : Type v))
            x))

Tactic:
ext W hW
-- 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)
U V : (OpenNhds x)ᵒᵖ
i : U ⟶ V
W : Opens (↑Y : Type v)
hW : unop (op ((OpenNhds.inclusion x).obj (unop U))) ≤ (Opens.map f).obj W
h :
  NatTrans.app (NatTrans.app (Adjunction.unit (pullbackPushforwardAdjunction C f)) F) (op W) ≫
      (obj (pullback C f) F).map (homOfLE hW).op ≫
        (obj (pullback C f) F).map (⋯.functor.map i.unop).op ≫
          germToPullbackStalk C f F (↑(unop V) : Opens (↑X : Type v)) x ⋯ =
    NatTrans.app (NatTrans.app (Adjunction.unit (pullbackPushforwardAdjunction C f)) F) (op W) ≫
      (obj (pullback C f) F).map (homOfLE hW).op ≫
        germToPullbackStalk C f F (↑(unop U) : Opens (↑X : Type v)) x ⋯ ≫
          𝟙
            (F.stalk
              ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := carrier)
                    f :
                  (a : (↑X : Type v)) → (↑Y : Type v))
                x))
⊢ NatTrans.app (NatTrans.app (Adjunction.unit (pullbackPushforwardAdjunction C f)) F) (op W) ≫
      (obj (pullback C f) F).map (homOfLE hW).op ≫
        (obj (pullback C f) F).map ((OpenNhds.inclusion x).map i.unop).op ≫
          germToPullbackStalk C f F (↑(unop V) : Opens (↑X : Type v)) x ⋯ =
    NatTrans.app (NatTrans.app (Adjunction.unit (pullbackPushforwardAdjunction C f)) F) (op W) ≫
      (obj (pullback C f) F).map (homOfLE hW).op ≫
        germToPullbackStalk C f F (↑(unop U) : Opens (↑X : Type v)) x ⋯ ≫
          𝟙
            (F.stalk
              ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := carrier)
                    f :
                  (a : (↑X : Type v)) → (↑Y : Type v))
                x))

Tactic:
dsimp [OpenNhds.inclusion]

Premises used: 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 : X ⟶ Y
F : Presheaf C Y
x : (↑X : Type v)
U V : (OpenNhds x)ᵒᵖ
i : U ⟶ V
W : Opens (↑Y : Type v)
hW : unop (op ((OpenNhds.inclusion x).obj (unop U))) ≤ (Opens.map f).obj W
h :
  NatTrans.app (NatTrans.app (Adjunction.unit (pullbackPushforwardAdjunction C f)) F) (op W) ≫
      (obj (pullback C f) F).map ((homOfLE hW).op ≫ (⋯.functor.map i.unop).op) ≫
        germToPullbackStalk C f F (↑(unop V) : Opens (↑X : Type v)) x ⋯ =
    F.germ W
      ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := carrier) f :
          (a : (↑X : Type v)) → (↑Y : Type v))
        x)
      ⋯
⊢ NatTrans.app (NatTrans.app (Adjunction.unit (pullbackPushforwardAdjunction C f)) F) (op W) ≫
      (obj (pullback C f) F).map (homOfLE hW).op ≫
        (obj (pullback C f) F).map (⋯.functor.map i.unop).op ≫
          germToPullbackStalk C f F (↑(unop V) : Opens (↑X : Type v)) x ⋯ =
    NatTrans.app (NatTrans.app (Adjunction.unit (pullbackPushforwardAdjunction C f)) F) (op W) ≫
      (obj (pullback C f) F).map (homOfLE hW).op ≫
        germToPullbackStalk C f F (↑(unop U) : Opens (↑X : Type v)) x ⋯ ≫
          𝟙
            (F.stalk
              ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := carrier)
                    f :
                  (a : (↑X : Type v)) → (↑Y : Type v))
                x))

Tactic:
rw [Category.comp_id, ← Functor.map_comp_assoc,
              pullbackPushforwardAdjunction_unit_pullback_map_germToPullbackStalk]

Premises used: CategoryTheory.Category.comp_id, TopCat.Presheaf.pullbackPushforwardAdjunction_unit_pullback_map_germToPullbackStalk, CategoryTheory.Functor.map_comp_assoc
-- 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)
U V : (OpenNhds x)ᵒᵖ
i : U ⟶ V
W : Opens (↑Y : Type v)
hW : unop (op ((OpenNhds.inclusion x).obj (unop U))) ≤ (Opens.map f).obj W
h h₁ :
  Eq (α :=
    F.obj (op W) ⟶
      F.stalk
        ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := carrier) f :
            (a : (↑X : Type v)) → (↑Y : Type v))
          x))
    (F.germ W
      ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := carrier) f :
          (a : (↑X : Type v)) → (↑Y : Type v))
        x)
      ⋯)
    (F.germ W
      ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := carrier) f :
          (a : (↑X : Type v)) → (↑Y : Type v))
        x)
      ⋯)
⊢ NatTrans.app (NatTrans.app (Adjunction.unit (pullbackPushforwardAdjunction C f)) F) (op W) ≫
      (obj (pullback C f) F).map ((homOfLE hW).op ≫ (⋯.functor.map i.unop).op) ≫
        germToPullbackStalk C f F (↑(unop V) : Opens (↑X : Type v)) x ⋯ =
    F.germ W
      ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := carrier) f :
          (a : (↑X : Type v)) → (↑Y : Type v))
        x)
      ⋯

Tactic:
erw [pullbackPushforwardAdjunction_unit_pullback_map_germToPullbackStalk]

Premises used: TopCat.Presheaf.pullbackPushforwardAdjunction_unit_pullback_map_germToPullbackStalk
-- 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 Y
x : (↑X : Type v)
V : Opens (↑X : Type v)
hV : x ∈ V
⊢ (obj (pullback C f) F).germ V x hV ≫ stalkPullbackInv C f F x = germToPullbackStalk C f F V x hV

Tactic:
apply colimit.ι_desc

Premises used: CategoryTheory.Limits.colimit.ι_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 : X ⟶ Y
F : Presheaf C Y
x : (↑X : Type v)
ih :
  ∀ (U : Opens (↑Y : Type v))
    (hU :
      Membership.mem (α := (↑Y : Type v)) U
        ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := carrier) f :
              (a : (↑X : Type v)) → (↑Y : Type v))
            x :
          (↑Y : Type v))),
    F.germ U
          ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := carrier) f :
              (a : (↑X : Type v)) → (↑Y : Type v))
            x)
          hU ≫
        stalkPullbackHom C f F x ≫ stalkPullbackInv C f F x =
      F.germ U
          ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := carrier) f :
              (a : (↑X : Type v)) → (↑Y : Type v))
            x)
          hU ≫
        𝟙
          (F.stalk
            ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := carrier) f :
                (a : (↑X : Type v)) → (↑Y : Type v))
              x))
⊢ stalkPullbackHom C f F x ≫ stalkPullbackInv C f F x =
    𝟙
      (F.stalk
        ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := carrier) f :
            (a : (↑X : Type v)) → (↑Y : Type v))
          x))

Tactic:
ext U hU
-- 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)
U : Opens (↑Y : Type v)
hU :
  Membership.mem (α := (↑Y : Type v)) U
    ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := carrier) f :
          (a : (↑X : Type v)) → (↑Y : Type v))
        x :
      (↑Y : Type v))
ih :
  F.germ U
        ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := carrier) f :
            (a : (↑X : Type v)) → (↑Y : Type v))
          x)
        hU ≫
      stalkPullbackHom C f F x ≫ stalkPullbackInv C f F x =
    F.germ U
        ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := carrier) f :
            (a : (↑X : Type v)) → (↑Y : Type v))
          x)
        hU ≫
      𝟙
        (F.stalk
          ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := carrier) f :
              (a : (↑X : Type v)) → (↑Y : Type v))
            x))
⊢ F.germ U
        ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := carrier) f :
            (a : (↑X : Type v)) → (↑Y : Type v))
          x)
        hU ≫
      stalkPullbackHom C f F x ≫ stalkPullbackInv C f F x =
    F.germ U
        ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := carrier) f :
            (a : (↑X : Type v)) → (↑Y : Type v))
          x)
        hU ≫
      𝟙
        (F.stalk
          ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := carrier) f :
              (a : (↑X : Type v)) → (↑Y : Type v))
            x))

Tactic:
dsimp