awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Subsheaf (goal)
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
F_1 F'_1 F'' : Functor.{v, w, u, w + 1} Cᵒᵖ (Type w)
G G' : Subfunctor F_1
F F' : Sheaf J (Type (max v u))
f : F ⟶ F'
I : Limits.MonoFactorisation f
h : Subfunctor.sheafify J (Subfunctor.range f.hom) ≤ Subfunctor.range I.m.hom
pf : IsIso (Subfunctor.toRange I.m.hom)
⊢ Subfunctor.homOfLe h ≫ inv (Subfunctor.toRange I.m.hom) ≫ I.m.hom =
    Subfunctor.homOfLe h ≫ (Subfunctor.range I.m.hom).ι
have h₁ : inv (Subfunctor.toRange I.m.hom) ≫ I.m.hom = (Subfunctor.range I.m.hom).ι
Tactic: congr 1
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Subsheaf (goal)
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
F_1 F'_1 F'' : Functor.{v, w, u, w + 1} Cᵒᵖ (Type w)
G G' : Subfunctor F_1
F F' : Sheaf J (Type (max v u))
f : F ⟶ F'
I : Limits.MonoFactorisation f
h : Subfunctor.sheafify J (Subfunctor.range f.hom) ≤ Subfunctor.range I.m.hom
pf : IsIso (Subfunctor.toRange I.m.hom)
⊢ inv (Subfunctor.toRange I.m.hom) ≫ I.m.hom = (Subfunctor.range I.m.hom).ι
have h₁ : I.m.hom = Subfunctor.toRange I.m.hom ≫ (Subfunctor.range I.m.hom).ι
Tactic: rw [IsIso.inv_comp_eq, Subfunctor.toRange_ι]
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Subsheaf (goal)
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
F_1 F'_1 F'' : Functor.{v, w, u, w + 1} Cᵒᵖ (Type w)
G G' : Subfunctor F_1
F F' : Sheaf J (Type (max v u))
f : F ⟶ F'
I : Limits.MonoFactorisation f
h : Subfunctor.sheafify J (Subfunctor.range f.hom) ≤ Subfunctor.range I.m.hom
pf : IsIso (Subfunctor.toRange I.m.hom)
h₁ : I.m.hom = Subfunctor.toRange I.m.hom ≫ (Subfunctor.range I.m.hom).ι
⊢ inv (Subfunctor.toRange I.m.hom) ≫ I.m.hom = (Subfunctor.range I.m.hom).ι
have h₂ : I.m.hom = I.m.hom
Tactic: rw [IsIso.inv_comp_eq, Subfunctor.toRange_ι]
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Types (goal)
Goal:
α : Type u
⊢ IsSheaf typesGrothendieckTopology (Functor.obj yoneda α)
have rw : Presieve.IsSheaf typesGrothendieckTopology (Functor.obj yoneda α)
Tactic: rw [isSheaf_iff_isSheaf_of_type]
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Types (goal)
Goal:
S : Type uᵒᵖ ⥤ Type u
hs : IsSheaf typesGrothendieckTopology S
α : Type u
f : α → S.obj (op PUnit.{u + 1})
⊢ (⇑(ConcreteCategory.hom (FC := TypeCat.Fun) (eval S α (typesGlue S hs α f))) :
      (a : (fun (X : Type u) ↦ X) α) → (fun (X : Type u) ↦ X) (S.obj (op PUnit.{u + 1}))) =
    f
have h : ∀ (x : (fun (X : Type u) ↦ X) α),
  (ConcreteCategory.hom (FC := TypeCat.Fun) (eval S α (typesGlue S hs α f)) :
        (a : (fun (X : Type u) ↦ X) α) → (fun (X : Type u) ↦ X) (S.obj (op PUnit.{u + 1})))
      x =
    f x
Tactic: funext x
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Types (goal)
Goal:
S : Type uᵒᵖ ⥤ Type u
hs : IsSheaf typesGrothendieckTopology S
α : Type u
f : α → S.obj (op PUnit.{u + 1})
x : (fun (X : Type u) ↦ X) α
⊢ (ConcreteCategory.hom (FC := TypeCat.Fun) (eval S α (typesGlue S hs α f)) :
        (a : (fun (X : Type u) ↦ X) α) → (fun (X : Type u) ↦ X) (S.obj (op PUnit.{u + 1})))
      x =
    f x
have h : (ConcreteCategory.hom (FC := TypeCat.Fun) (S.map (Quiver.Hom.op (↾fun (x : PUnit.{u + 1}) ↦ PUnit.unit))) :
      (a : (fun (X : Type u) ↦ X) (S.obj (op PUnit.{u + 1}))) → (fun (X : Type u) ↦ X) (S.obj (op PUnit.{u + 1})))
    (f
      ((ConcreteCategory.hom (FC := TypeCat.Fun) (↾fun (x_1 : PUnit.{u + 1}) ↦ x) :
          (a : (fun (X : Type u) ↦ X) PUnit.{u + 1}) → (fun (X : Type u) ↦ X) α)
        (Classical.choose (p := fun (x : PUnit.{u + 1}) ↦ ∀ (y : PUnit.{u + 1}), y = x) sorry))) =
  f x
Tactic: apply (IsSheafFor.valid_glue _ _ _ <| ⟨PUnit.unit, fun _ => Subsingleton.elim _ _⟩).trans
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Types (goal)
Goal:
S : Type uᵒᵖ ⥤ Type u
hs : IsSheaf typesGrothendieckTopology S
α : Type u
s : S.obj (op α)
⊢ typesGlue S hs α
      (⇑(ConcreteCategory.hom (FC := TypeCat.Fun) (eval S α s)) :
        (a : (fun (X : Type u) ↦ X) α) → (fun (X : Type u) ↦ X) (S.obj (op PUnit.{u + 1}))) =
    s
have apply : ∀ ⦃Y : Type u⦄ ⦃f : Y ⟶ α⦄,
  discretePresieve α f →
    Eq (α := (fun (X : Type u) ↦ X) (S.obj (op Y)))
      ((ConcreteCategory.hom (FC := TypeCat.Fun) (S.map f.op) :
            (a : (fun (X : Type u) ↦ X) (S.obj (op α))) → (fun (X : Type u) ↦ X) (S.obj (op Y)))
          (typesGlue S hs α
            (⇑(ConcreteCategory.hom (FC := TypeCat.Fun) (eval S α s)) :
              (a : (fun (X : Type u) ↦ X) α) → (fun (X : Type u) ↦ X) (S.obj (op PUnit.{u + 1})))) :
        (fun (X : Type u) ↦ X) (S.obj (op Y)))
      ((ConcreteCategory.hom (FC := TypeCat.Fun) (S.map f.op) :
            (a : (fun (X : Type u) ↦ X) (S.obj (op α))) → (fun (X : Type u) ↦ X) (S.obj (op Y)))
          s :
        (fun (X : Type u) ↦ X) (S.obj (op Y)))
Tactic: apply (hs.isSheafFor _ (generate_discretePresieve_mem α)).isSeparatedFor.ext
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Types (goal)
Goal:
S : Type uᵒᵖ ⥤ Type u
hs : IsSheaf typesGrothendieckTopology S
α : Type u
s : S.obj (op α)
β : Type u
f : β ⟶ α
hf : discretePresieve α f
⊢ Eq (α := (fun (X : Type u) ↦ X) (S.obj (op β)))
    ((ConcreteCategory.hom (FC := TypeCat.Fun) (S.map f.op) :
          (a : (fun (X : Type u) ↦ X) (S.obj (op α))) → (fun (X : Type u) ↦ X) (S.obj (op β)))
        (typesGlue S hs α
          (⇑(ConcreteCategory.hom (FC := TypeCat.Fun) (eval S α s)) :
            (a : (fun (X : Type u) ↦ X) α) → (fun (X : Type u) ↦ X) (S.obj (op PUnit.{u + 1})))) :
      (fun (X : Type u) ↦ X) (S.obj (op β)))
    ((ConcreteCategory.hom (FC := TypeCat.Fun) (S.map f.op) :
          (a : (fun (X : Type u) ↦ X) (S.obj (op α))) → (fun (X : Type u) ↦ X) (S.obj (op β)))
        s :
      (fun (X : Type u) ↦ X) (S.obj (op β)))
have apply : Eq (α := (fun (X : Type u) ↦ X) (S.obj (op β)))
  ((ConcreteCategory.hom (FC := TypeCat.Fun) (S.map (Quiver.Hom.op (↾fun (x : β) ↦ PUnit.unit))) :
        (a : (fun (X : Type u) ↦ X) (S.obj (op PUnit.{u + 1}))) → (fun (X : Type u) ↦ X) (S.obj (op β)))
      ((ConcreteCategory.hom (FC := TypeCat.Fun) (eval S α s) :
            (a : (fun (X : Type u) ↦ X) α) → (fun (X : Type u) ↦ X) (S.obj (op PUnit.{u + 1})))
          ((ConcreteCategory.hom (FC := TypeCat.Fun) f : (a : (fun (X : Type u) ↦ X) β) → (fun (X : Type u) ↦ X) α)
              (Classical.choose (p := fun (x : β) ↦ ∀ (y : β), y = x) hf) :
            (fun (X : Type u) ↦ X) α) :
        (fun (X : Type u) ↦ X) (S.obj (op PUnit.{u + 1}))) :
    (fun (X : Type u) ↦ X) (S.obj (op β)))
  ((ConcreteCategory.hom (FC := TypeCat.Fun) (S.map f.op) :
        (a : (fun (X : Type u) ↦ X) (S.obj (op α))) → (fun (X : Type u) ↦ X) (S.obj (op β)))
      s :
    (fun (X : Type u) ↦ X) (S.obj (op β)))
Tactic: apply (IsSheafFor.valid_glue _ _ _ hf).trans
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Types (goal)
Goal:
S : Type uᵒᵖ ⥤ Type u
hs : IsSheaf typesGrothendieckTopology S
α : Type u
s : S.obj (op α)
β : Type u
f : β ⟶ α
hf : discretePresieve α f
⊢ Eq (α := (fun (X : Type u) ↦ X) (S.obj (op β)))
    ((ConcreteCategory.hom (FC := TypeCat.Fun) (S.map (Quiver.Hom.op (↾fun (x : β) ↦ PUnit.unit))) :
          (a : (fun (X : Type u) ↦ X) (S.obj (op PUnit.{u + 1}))) → (fun (X : Type u) ↦ X) (S.obj (op β)))
        ((ConcreteCategory.hom (FC := TypeCat.Fun) (eval S α s) :
              (a : (fun (X : Type u) ↦ X) α) → (fun (X : Type u) ↦ X) (S.obj (op PUnit.{u + 1})))
            ((ConcreteCategory.hom (FC := TypeCat.Fun) f : (a : (fun (X : Type u) ↦ X) β) → (fun (X : Type u) ↦ X) α)
                (Classical.choose (p := fun (x : β) ↦ ∀ (y : β), y = x) hf) :
              (fun (X : Type u) ↦ X) α) :
          (fun (X : Type u) ↦ X) (S.obj (op PUnit.{u + 1}))) :
      (fun (X : Type u) ↦ X) (S.obj (op β)))
    ((ConcreteCategory.hom (FC := TypeCat.Fun) (S.map f.op) :
          (a : (fun (X : Type u) ↦ X) (S.obj (op α))) → (fun (X : Type u) ↦ X) (S.obj (op β)))
        s :
      (fun (X : Type u) ↦ X) (S.obj (op β)))
have simp : Eq (α := S.obj (op β))
  ((ConcreteCategory.hom (FC := TypeCat.Fun)
          (S.map
            (Quiver.Hom.op
              (CategoryStruct.comp (Y := PUnit.{u + 1}) (↾fun (x : β) ↦ PUnit.unit)
                (↾fun (x : PUnit.{u + 1}) ↦
                  (ConcreteCategory.hom (FC := TypeCat.Fun) f : (a : β) → α)
                    (Classical.choose (p := fun (x : β) ↦ ∀ (y : β), y = x) hf))))) :
        (a : S.obj (op α)) → S.obj (op β))
      s :
    S.obj (op β))
  ((ConcreteCategory.hom (FC := TypeCat.Fun) (S.map f.op) : (a : S.obj (op α)) → S.obj (op β)) s : S.obj (op β))
Tactic: simp only [eval, ConcreteCategory.hom_ofHom, TypeCat.Fun.coe_mk, ← comp_apply,
    ← Functor.map_comp, ← op_comp]
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Types (goal)
Goal:
S : Type uᵒᵖ ⥤ Type u
hs : IsSheaf typesGrothendieckTopology S
α : Type u
s : S.obj (op α)
β : Type u
f : β ⟶ α
hf : discretePresieve α f
⊢ Eq (α := S.obj (op β))
    ((ConcreteCategory.hom (FC := TypeCat.Fun)
            (S.map
              (Quiver.Hom.op
                (CategoryStruct.comp (Y := PUnit.{u + 1}) (↾fun (x : β) ↦ PUnit.unit)
                  (↾fun (x : PUnit.{u + 1}) ↦
                    (ConcreteCategory.hom (FC := TypeCat.Fun) f : (a : β) → α)
                      (Classical.choose (p := fun (x : β) ↦ ∀ (y : β), y = x) hf))))) :
          (a : S.obj (op α)) → S.obj (op β))
        s :
      S.obj (op β))
    ((ConcreteCategory.hom (FC := TypeCat.Fun) (S.map f.op) : (a : S.obj (op α)) → S.obj (op β)) s : S.obj (op β))
have e_a : CategoryStruct.comp (Y := PUnit.{u + 1}) (↾fun (x : β) ↦ PUnit.unit)
    (↾fun (x : PUnit.{u + 1}) ↦
      (ConcreteCategory.hom (FC := TypeCat.Fun) f : (a : β) → α)
        (Classical.choose (p := fun (x : β) ↦ ∀ (y : β), y = x) hf)) =
  f
Tactic: congr
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Types (goal)
Goal:
S : Type uᵒᵖ ⥤ Type u
hs : IsSheaf typesGrothendieckTopology S
α : Type u
s : S.obj (op α)
β : Type u
f : β ⟶ α
hf : discretePresieve α f
⊢ CategoryStruct.comp (Y := PUnit.{u + 1}) (↾fun (x : β) ↦ PUnit.unit)
      (↾fun (x : PUnit.{u + 1}) ↦
        (ConcreteCategory.hom (FC := TypeCat.Fun) f : (a : β) → α)
          (Classical.choose (p := fun (x : β) ↦ ∀ (y : β), y = x) hf)) =
    f
have e_a : ∀ (x : β),
  TypeCat.Fun.toFun
      (ConcreteCategory.hom (FC := TypeCat.Fun)
        (CategoryStruct.comp (Y := PUnit.{u + 1}) (↾fun (x : β) ↦ PUnit.unit)
          (↾fun (x : PUnit.{u + 1}) ↦
            (ConcreteCategory.hom (FC := TypeCat.Fun) f : (a : β) → α)
              (Classical.choose (p := fun (x : β) ↦ ∀ (y : β), y = x) hf))))
      x =
    (ConcreteCategory.hom (FC := TypeCat.Fun) f).toFun x
Tactic: ext x
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Types (goal)
Goal:
S : Type uᵒᵖ ⥤ Type u
hs : Presheaf.IsSheaf typesGrothendieckTopology S
α : Type u
x : α ⟶ S.obj (op PUnit.{u + 1})
⊢ eval S α
      ((fun (f : α ⟶ S.obj (op PUnit.{u + 1})) ↦
          typesGlue S ⋯ α
            (⇑(ConcreteCategory.hom (FC := TypeCat.Fun) f) :
              (a : (fun (X : Type u) ↦ X) α) → (fun (X : Type u) ↦ X) (S.obj (op PUnit.{u + 1}))))
        x) =
    x
have h : ∀ (x_1 : α),
  (ConcreteCategory.hom (FC := TypeCat.Fun)
          (eval S α
            ((fun (f : α ⟶ S.obj (op PUnit.{u + 1})) ↦
                typesGlue S sorry α
                  (⇑(ConcreteCategory.hom (FC := TypeCat.Fun) f) :
                    (a : (fun (X : Type u) ↦ X) α) → (fun (X : Type u) ↦ X) (S.obj (op PUnit.{u + 1}))))
              x))).toFun
      x_1 =
    (ConcreteCategory.hom (FC := TypeCat.Fun) x).toFun x_1
Tactic: ext
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Types (goal)
Goal:
S : Type uᵒᵖ ⥤ Type u
α β : Type u
f : β ⟶ α
s : (fun (X : Type u) ↦ X) (S.obj (op α))
x : (fun (X : Type u) ↦ X) β
⊢ Eq (α := (fun (X : Type u) ↦ X) (S.obj (op PUnit.{u + 1})))
    ((ConcreteCategory.hom (FC := TypeCat.Fun)
            (eval S β
              ((ConcreteCategory.hom (FC := TypeCat.Fun) (S.map f.op) :
                  (a : (fun (X : Type u) ↦ X) (S.obj (op α))) → (fun (X : Type u) ↦ X) (S.obj (op β)))
                s)) :
          (a : (fun (X : Type u) ↦ X) β) → (fun (X : Type u) ↦ X) (S.obj (op PUnit.{u + 1})))
        x :
      (fun (X : Type u) ↦ X) (S.obj (op PUnit.{u + 1})))
    ((ConcreteCategory.hom (FC := TypeCat.Fun) (eval S α s) :
          (a : (fun (X : Type u) ↦ X) α) → (fun (X : Type u) ↦ X) (S.obj (op PUnit.{u + 1})))
        ((ConcreteCategory.hom (FC := TypeCat.Fun) f : (a : (fun (X : Type u) ↦ X) β) → (fun (X : Type u) ↦ X) α) x :
          (fun (X : Type u) ↦ X) α) :
      (fun (X : Type u) ↦ X) (S.obj (op PUnit.{u + 1})))
have simp_rw : Eq (α := S.obj (op PUnit.{u + 1}))
  ((ConcreteCategory.hom (FC := TypeCat.Fun)
          (↾fun (x : β) ↦
            (ConcreteCategory.hom (FC := TypeCat.Fun) (S.map (Quiver.Hom.op ((↾fun (x_1 : PUnit.{u + 1}) ↦ x) ≫ f))) :
                (a : S.obj (op α)) → S.obj (op PUnit.{u + 1}))
              s) :
        (a : β) → S.obj (op PUnit.{u + 1}))
      x :
    S.obj (op PUnit.{u + 1}))
  ((ConcreteCategory.hom (FC := TypeCat.Fun)
          (f ≫
            ↾fun (x : α) ↦
              (ConcreteCategory.hom (FC := TypeCat.Fun) (S.map (Quiver.Hom.op (↾fun (x_1 : PUnit.{u + 1}) ↦ x))) :
                  (a : S.obj (op α)) → S.obj (op PUnit.{u + 1}))
                s) :
        (a : β) → S.obj (op PUnit.{u + 1}))
      x :
    S.obj (op PUnit.{u + 1}))
Tactic: simp_rw [eval, ← comp_apply, ← Functor.map_comp, ← op_comp]
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Types (goal)
Goal:
S : Type uᵒᵖ ⥤ Type u
hs : Presheaf.IsSheaf typesGrothendieckTopology S
α β : Type uᵒᵖ
f : α ⟶ β
⊢ Eq (α := S.obj α ⟶ Functor.obj (Functor.obj yoneda (S.obj (op PUnit.{u + 1}))) β)
    (S.map f ≫
      Iso.hom
        ((fun (α : Type uᵒᵖ) ↦
            (Equiv.toIso (evalEquiv S hs (unop α)) : S.obj (op (unop α)) ≅ unop α ⟶ S.obj (op PUnit.{u + 1})))
          β))
    (Iso.hom
        ((fun (α : Type uᵒᵖ) ↦
            (Equiv.toIso (evalEquiv S hs (unop α)) : S.obj (op (unop α)) ≅ unop α ⟶ S.obj (op PUnit.{u + 1})))
          α) ≫
      Functor.map (Functor.obj yoneda (S.obj (op PUnit.{u + 1}))) f)
have dsimp : S.map f ≫ Iso.hom (Equiv.toIso (evalEquiv S hs (unop β))) =
  Iso.hom (Equiv.toIso (evalEquiv S hs (unop α))) ≫ ↾fun (g : unop α ⟶ S.obj (op PUnit.{u + 1})) ↦ f.unop ≫ g
Tactic: dsimp
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Types (goal)
Goal:
S : Type uᵒᵖ ⥤ Type u
hs : Presheaf.IsSheaf typesGrothendieckTopology S
α β : Type uᵒᵖ
f : α ⟶ β
⊢ S.map f ≫ Iso.hom (Equiv.toIso (evalEquiv S hs (unop β))) =
    Iso.hom (Equiv.toIso (evalEquiv S hs (unop α))) ≫ ↾fun (g : unop α ⟶ S.obj (op PUnit.{u + 1})) ↦ f.unop ≫ g
have h : ∀ (x : S.obj α) (x_1 : unop β),
  (ConcreteCategory.hom (FC := TypeCat.Fun)
          ((ConcreteCategory.hom (FC := TypeCat.Fun) (S.map f ≫ Iso.hom (Equiv.toIso (evalEquiv S hs (unop β))))).toFun
            x)).toFun
      x_1 =
    TypeCat.Fun.toFun
      (@ConcreteCategory.hom _ types TypeCat.Fun _ _ _ _ _
        (TypeCat.Fun.toFun
          (ConcreteCategory.hom (FC := TypeCat.Fun)
            (Iso.hom (Equiv.toIso (evalEquiv S hs (unop α))) ≫
              ↾fun (g : unop α ⟶ S.obj (op PUnit.{u + 1})) ↦ f.unop ≫ g))
          x))
      x_1
Tactic: ext
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Types (goal)
Goal:
S : Sheaf typesGrothendieckTopology (Type u)
⊢ CategoryStruct.comp (Y :=
      yoneda'.obj (S.obj (P := Presheaf.IsSheaf typesGrothendieckTopology).obj (op PUnit.{u + 1})))
      { hom := (equivYoneda (S.obj (P := Presheaf.IsSheaf typesGrothendieckTopology)) ⋯).hom }
      { hom := (equivYoneda (S.obj (P := Presheaf.IsSheaf typesGrothendieckTopology)) ⋯).inv } =
    𝟙 S
have h : InducedCategory.Hom.hom
    (CategoryStruct.comp (Y :=
      yoneda'.obj (S.obj (P := Presheaf.IsSheaf typesGrothendieckTopology).obj (op PUnit.{u + 1})))
      { hom := (equivYoneda (S.obj (P := Presheaf.IsSheaf typesGrothendieckTopology)) sorry).hom }
      { hom := (equivYoneda (S.obj (P := Presheaf.IsSheaf typesGrothendieckTopology)) sorry).inv }) =
  InducedCategory.Hom.hom (𝟙 S)
Tactic: ext1
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Types (goal)
Goal:
S : Sheaf typesGrothendieckTopology (Type u)
⊢ CategoryStruct.comp (Y := S) { hom := (equivYoneda (S.obj (P := Presheaf.IsSheaf typesGrothendieckTopology)) ⋯).inv }
      { hom := (equivYoneda (S.obj (P := Presheaf.IsSheaf typesGrothendieckTopology)) ⋯).hom } =
    𝟙 (yoneda'.obj (S.obj (P := Presheaf.IsSheaf typesGrothendieckTopology).obj (op PUnit.{u + 1})))
have h : InducedCategory.Hom.hom
    (CategoryStruct.comp (Y := S)
      { hom := (equivYoneda (S.obj (P := Presheaf.IsSheaf typesGrothendieckTopology)) sorry).inv }
      { hom := (equivYoneda (S.obj (P := Presheaf.IsSheaf typesGrothendieckTopology)) sorry).hom }) =
  InducedCategory.Hom.hom
    (𝟙 (yoneda'.obj (S.obj (P := Presheaf.IsSheaf typesGrothendieckTopology).obj (op PUnit.{u + 1}))))
Tactic: ext1
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Types (goal)
Goal:
S₁ S₂ : Sheaf typesGrothendieckTopology (Type u)
f : S₁ ⟶ S₂
⊢ Eq (α :=
    Functor.obj (𝟭 (Sheaf typesGrothendieckTopology (Type u))) S₁ ⟶
      ((sheafToPresheaf typesGrothendieckTopology (Type u) ⋙
              Functor.obj (evaluation Type uᵒᵖ (Type u)) (op PUnit.{u + 1})) ⋙
            yoneda').obj
        S₂)
    (Functor.map (𝟭 (Sheaf typesGrothendieckTopology (Type u))) f ≫
      Iso.hom ((fun (S : Sheaf typesGrothendieckTopology (Type u)) ↦ equivYoneda' S) S₂))
    (Iso.hom ((fun (S : Sheaf typesGrothendieckTopology (Type u)) ↦ equivYoneda' S) S₁) ≫
      ((sheafToPresheaf typesGrothendieckTopology (Type u) ⋙
              Functor.obj (evaluation Type uᵒᵖ (Type u)) (op PUnit.{u + 1})) ⋙
            yoneda').map
        f)
have h : ∀ (α : Type u)
  (s :
    (Functor.obj (𝟭 (Sheaf typesGrothendieckTopology (Type u))) S₁).obj (P :=
          Presheaf.IsSheaf typesGrothendieckTopology).obj
      (op α)),
  TypeCat.Fun.toFun
      (ConcreteCategory.hom (FC := TypeCat.Fun)
        (NatTrans.app
          (InducedCategory.Hom.hom
            (Functor.map (𝟭 (Sheaf typesGrothendieckTopology (Type u))) f ≫
              Iso.hom ((fun (S : Sheaf typesGrothendieckTopology (Type u)) ↦ equivYoneda' S) S₂)))
          (op α)))
      s =
    TypeCat.Fun.toFun
      (ConcreteCategory.hom (FC := TypeCat.Fun)
        (NatTrans.app
          (InducedCategory.Hom.hom
            (Iso.hom ((fun (S : Sheaf typesGrothendieckTopology (Type u)) ↦ equivYoneda' S) S₁) ≫
              ((sheafToPresheaf typesGrothendieckTopology (Type u) ⋙
                      Functor.obj (evaluation Type uᵒᵖ (Type u)) (op PUnit.{u + 1})) ⋙
                    yoneda').map
                f))
          (op α)))
      s
Tactic: ext ⟨α⟩ s
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Types (goal)
Goal:
S₁ S₂ : Sheaf typesGrothendieckTopology (Type u)
f : S₁ ⟶ S₂
α : Type u
s :
  (Functor.obj (𝟭 (Sheaf typesGrothendieckTopology (Type u))) S₁).obj (P :=
        Presheaf.IsSheaf typesGrothendieckTopology).obj
    (op α)
⊢ TypeCat.Fun.toFun
      (ConcreteCategory.hom (FC := TypeCat.Fun)
        (NatTrans.app
          (InducedCategory.Hom.hom
            (Functor.map (𝟭 (Sheaf typesGrothendieckTopology (Type u))) f ≫
              Iso.hom ((fun (S : Sheaf typesGrothendieckTopology (Type u)) ↦ equivYoneda' S) S₂)))
          (op α)))
      s =
    TypeCat.Fun.toFun
      (ConcreteCategory.hom (FC := TypeCat.Fun)
        (NatTrans.app
          (InducedCategory.Hom.hom
            (Iso.hom ((fun (S : Sheaf typesGrothendieckTopology (Type u)) ↦ equivYoneda' S) S₁) ≫
              ((sheafToPresheaf typesGrothendieckTopology (Type u) ⋙
                      Functor.obj (evaluation Type uᵒᵖ (Type u)) (op PUnit.{u + 1})) ⋙
                    yoneda').map
                f))
          (op α)))
      s
have h : Eq (α := α ⟶ S₂.obj (P := Presheaf.IsSheaf typesGrothendieckTopology).obj (op PUnit.{u + 1}))
  ((ConcreteCategory.hom (FC := TypeCat.Fun)
          ((equivYoneda (S₂.obj (P := Presheaf.IsSheaf typesGrothendieckTopology)) sorry).hom.app (op α)) :
        (a : S₂.obj.obj (op α)) → α ⟶ S₂.obj.obj (op PUnit.{u + 1}))
      ((ConcreteCategory.hom (FC := TypeCat.Fun) (f.hom.app (op α)) : (a : S₁.obj.obj (op α)) → S₂.obj.obj (op α)) s :
        S₂.obj (P := Presheaf.IsSheaf typesGrothendieckTopology).obj (op α)) :
    α ⟶ S₂.obj (P := Presheaf.IsSheaf typesGrothendieckTopology).obj (op PUnit.{u + 1}))
  ((ConcreteCategory.hom (FC := TypeCat.Fun)
            ((equivYoneda (S₁.obj (P := Presheaf.IsSheaf typesGrothendieckTopology)) sorry).hom.app (op α)) :
          (a : S₁.obj.obj (op α)) → α ⟶ S₁.obj.obj (op PUnit.{u + 1}))
        s ≫
      f.hom.app (op PUnit.{u + 1}) :
    α ⟶ S₂.obj (P := Presheaf.IsSheaf typesGrothendieckTopology).obj (op PUnit.{u + 1}))
Tactic: dsimp at s ⊢
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Types (goal)
Goal:
S₁ S₂ : Sheaf typesGrothendieckTopology (Type u)
f : S₁ ⟶ S₂
α : Type u
s : S₁.obj (P := Presheaf.IsSheaf typesGrothendieckTopology).obj (op α)
⊢ Eq (α := α ⟶ S₂.obj (P := Presheaf.IsSheaf typesGrothendieckTopology).obj (op PUnit.{u + 1}))
    ((ConcreteCategory.hom (FC := TypeCat.Fun)
            ((equivYoneda (S₂.obj (P := Presheaf.IsSheaf typesGrothendieckTopology)) ⋯).hom.app (op α)) :
          (a : S₂.obj.obj (op α)) → α ⟶ S₂.obj.obj (op PUnit.{u + 1}))
        ((ConcreteCategory.hom (FC := TypeCat.Fun) (f.hom.app (op α)) : (a : S₁.obj.obj (op α)) → S₂.obj.obj (op α)) s :
          S₂.obj (P := Presheaf.IsSheaf typesGrothendieckTopology).obj (op α)) :
      α ⟶ S₂.obj (P := Presheaf.IsSheaf typesGrothendieckTopology).obj (op PUnit.{u + 1}))
    ((ConcreteCategory.hom (FC := TypeCat.Fun)
              ((equivYoneda (S₁.obj (P := Presheaf.IsSheaf typesGrothendieckTopology)) ⋯).hom.app (op α)) :
            (a : S₁.obj.obj (op α)) → α ⟶ S₁.obj.obj (op PUnit.{u + 1}))
          s ≫
        f.hom.app (op PUnit.{u + 1}) :
      α ⟶ S₂.obj (P := Presheaf.IsSheaf typesGrothendieckTopology).obj (op PUnit.{u + 1}))
have h : ∀ (x : α),
  Eq (α := S₂.obj (P := Presheaf.IsSheaf typesGrothendieckTopology).obj (op PUnit.{u + 1}))
    (TypeCat.Fun.toFun
      (@ConcreteCategory.hom _ types TypeCat.Fun _ _ _ _ _
        ((ConcreteCategory.hom (FC := TypeCat.Fun)
              ((equivYoneda (S₂.obj (P := Presheaf.IsSheaf typesGrothendieckTopology)) sorry).hom.app (op α)) :
            (a : S₂.obj.obj (op α)) → α ⟶ S₂.obj.obj (op PUnit.{u + 1}))
          ((ConcreteCategory.hom (FC := TypeCat.Fun) (f.hom.app (op α)) : (a : S₁.obj.obj (op α)) → S₂.obj.obj (op α))
              s :
            S₂.obj (P := Presheaf.IsSheaf typesGrothendieckTopology).obj (op α))))
      x)
    (TypeCat.Fun.toFun
      (ConcreteCategory.hom (FC := TypeCat.Fun)
        ((ConcreteCategory.hom (FC := TypeCat.Fun)
                ((equivYoneda (S₁.obj (P := Presheaf.IsSheaf typesGrothendieckTopology)) sorry).hom.app (op α)) :
              (a : S₁.obj.obj (op α)) → α ⟶ S₁.obj.obj (op PUnit.{u + 1}))
            s ≫
          f.hom.app (op PUnit.{u + 1})))
      x)
Tactic: ext x