awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.CategoryTheory.Comma.Presheaf.Basic (goal)
Goal:
C : Type u
inst : Category.{v, u} C
A F G : Functor.{v, v, u, v + 1} Cᵒᵖ (Type v)
η : F ⟶ A
μ : G ⟶ A
ε : F ⟶ G
hε : ε ≫ μ = η
X : C
s : Functor.obj yoneda X ⟶ A
u : F.obj (op X)
h : MakesOverArrow η s u
rw :
  Eq (α := (fun (X : Type v) ↦ X) (A.obj (op X)))
    ((ConcreteCategory.hom (FC := TypeCat.Fun) (ε.app (op X) ≫ μ.app (op X)) :
          (a : (fun (X : Type v) ↦ X) (F.obj (op X))) → A.obj (op X))
        u :
      A.obj (op X))
    ((yonedaEquiv : (a : yoneda.obj X ⟶ A) → A.obj (op X)) s : A.obj (op X))
rw₁ :
  Eq (α := (fun (X : Type v) ↦ X) (A.obj (op X)))
    ((ConcreteCategory.hom (FC := TypeCat.Fun) ((ε ≫ μ).app (op X)) :
          (a : (fun (X : Type v) ↦ X) (F.obj (op X))) → A.obj (op X))
        u :
      A.obj (op X))
    ((yonedaEquiv : (a : yoneda.obj X ⟶ A) → A.obj (op X)) s : A.obj (op X))
rw₂ :
  Eq (α := (fun (X : Type v) ↦ X) (A.obj (op X)))
    ((ConcreteCategory.hom (FC := TypeCat.Fun) (η.app (op X)) :
          (a : (fun (X : Type v) ↦ X) (F.obj (op X))) → A.obj (op X))
        u :
      A.obj (op X))
    ((yonedaEquiv : (a : yoneda.obj X ⟶ A) → A.obj (op X)) s : A.obj (op X))
⊢ Eq (α := (fun (X : Type v) ↦ X) (A.obj (op X)))
    ((ConcreteCategory.hom (FC := TypeCat.Fun) (μ.app (op X)) :
          (a : (fun (X : Type v) ↦ X) (G.obj (op X))) → (fun (X : Type v) ↦ X) (A.obj (op X)))
        ((ConcreteCategory.hom (FC := TypeCat.Fun) (ε.app (op X)) :
              (a : (fun (X : Type v) ↦ X) (F.obj (op X))) → (fun (X : Type v) ↦ X) (G.obj (op X)))
            u :
          (fun (X : Type v) ↦ X) (G.obj (op X))) :
      (fun (X : Type v) ↦ X) (A.obj (op X)))
    ((yonedaEquiv : (a : yoneda.obj X ⟶ A) → A.obj (op X)) s : A.obj (op X))
have rw₃ : Eq.{v + 1} (α := (fun (X : Type v) ↦ X) (A.obj (op X)))
  ((yonedaEquiv : (a : yoneda.obj X ⟶ A) → A.obj (op X)) s : A.obj (op X))
  ((yonedaEquiv : (a : yoneda.obj X ⟶ A) → A.obj (op X)) s : A.obj (op X))
Tactic: rw [← comp_apply, ← NatTrans.comp_app, hε, h.app]
-- Lean 4 proof step from Mathlib.CategoryTheory.Comma.Presheaf.Basic (goal)
Goal:
C : Type u
inst : Category.{v, u} C
A : Functor.{v, v, u, v + 1} Cᵒᵖ (Type v)
Y : C
η : Functor.obj yoneda Y ⟶ A
X : C
s : Functor.obj yoneda X ⟶ A
p : OverArrows η s
⊢ Functor.map yoneda p.val ≫ η = s
have rw : Eq (α := A.obj (op (unop (op X))))
  ((ConcreteCategory.hom (FC := TypeCat.Fun) (η.app (op (unop (op X)))) :
        (a : (fun (X : Type v) ↦ X) ((yoneda.obj Y).obj (op (unop (op X))))) →
          (fun (X : Type v) ↦ X) (A.obj (op (unop (op X)))))
      p.val :
    (fun (X : Type v) ↦ X) (A.obj (op (unop (op X)))))
  ((yonedaEquiv : (a : yoneda.obj (unop (op X)) ⟶ A) → A.obj (op (unop (op X)))) s : A.obj (op (unop (op X))))
Tactic: rw [← yonedaEquiv.injective.eq_iff, yonedaEquiv_comp, yonedaEquiv_yoneda_map]
-- Lean 4 proof step from Mathlib.CategoryTheory.Comma.Presheaf.Basic (goal)
Goal:
C : Type u
inst : Category.{v, u} C
A : Functor.{v, v, u, v + 1} Cᵒᵖ (Type v)
F : (CostructuredArrow yoneda A)ᵒᵖ ⥤ Type v
X Y : C
f : X ⟶ Y
g g' : Functor.obj yoneda Y ⟶ A
h : g = g'
x : F.obj (op (CostructuredArrow.mk g'))
⊢ op (CostructuredArrow.mk g') = op (CostructuredArrow.mk g)
have rw : op (CostructuredArrow.mk g') = op (CostructuredArrow.mk g')
Tactic: rw [h]
-- Lean 4 proof step from Mathlib.CategoryTheory.Comma.Presheaf.Basic (goal)
Goal:
C : Type u
inst : Category.{v, u} C
A : Functor.{v, v, u, v + 1} Cᵒᵖ (Type v)
F : (CostructuredArrow yoneda A)ᵒᵖ ⥤ Type v
X Y : C
f : X ⟶ Y
g g' : Functor.obj yoneda Y ⟶ A
h : g = g'
x : F.obj (op (CostructuredArrow.mk g'))
⊢ op (CostructuredArrow.mk (Functor.map yoneda f ≫ g')) = op (CostructuredArrow.mk (Functor.map yoneda f ≫ g))
have rw : op (CostructuredArrow.mk (Functor.map yoneda f ≫ g')) = op (CostructuredArrow.mk (Functor.map yoneda f ≫ g'))
Tactic: rw [h]
-- Lean 4 proof step from Mathlib.CategoryTheory.Comma.Presheaf.Basic (goal)
Goal:
C : Type u
inst : Category.{v, u} C
A : Functor.{v, v, u, v + 1} Cᵒᵖ (Type v)
F : (CostructuredArrow yoneda A)ᵒᵖ ⥤ Type v
X : C
s : Functor.obj yoneda X ⟶ A
x : F.obj (op (CostructuredArrow.mk s))
⊢ op (CostructuredArrow.mk s) =
    op
      (CostructuredArrow.mk (Y := X)
        ((Equiv.symm yonedaEquiv : (a : A.obj (op X)) → yoneda.obj X ⟶ A)
          ((yonedaEquiv : (a : yoneda.obj X ⟶ A) → A.obj (op X)) s)))
have rw : op (CostructuredArrow.mk s) = op (CostructuredArrow.mk s)
Tactic: rw [Equiv.symm_apply_apply]
-- Lean 4 proof step from Mathlib.CategoryTheory.Comma.Presheaf.Basic (goal)
Goal:
C : Type u
inst : Category.{v, u} C
A : Functor.{v, v, u, v + 1} Cᵒᵖ (Type v)
F : (CostructuredArrow yoneda A)ᵒᵖ ⥤ Type v
X : C
s : Functor.obj yoneda X ⟶ A
x : F.obj (op (CostructuredArrow.mk s))
⊢ op (CostructuredArrow.mk s) = op (CostructuredArrow.mk (mk s x).fst)
have rw : op (CostructuredArrow.mk s) = op (CostructuredArrow.mk s)
Tactic: rw [YonedaCollection.mk_fst]
-- Lean 4 proof step from Mathlib.CategoryTheory.Comma.Presheaf.Basic (goal)
Goal:
C : Type u
inst : Category.{v, u} C
A : Functor.{v, v, u, v + 1} Cᵒᵖ (Type v)
F : (CostructuredArrow yoneda A)ᵒᵖ ⥤ Type v
X : C
p q : YonedaCollection F X
h : p.fst = q.fst
⊢ op (CostructuredArrow.mk q.fst) = op (CostructuredArrow.mk p.fst)
have rw : op (CostructuredArrow.mk q.fst) = op (CostructuredArrow.mk q.fst)
Tactic: rw [h]
-- Lean 4 proof step from Mathlib.CategoryTheory.Comma.Presheaf.Basic (goal)
Goal:
C : Type u
inst : Category.{v, u} C
A : Functor.{v, v, u, v + 1} Cᵒᵖ (Type v)
F : (CostructuredArrow yoneda A)ᵒᵖ ⥤ Type v
X : C
p q : YonedaCollection F X
h : p.fst = q.fst
h' :
  (ConcreteCategory.hom (FC := TypeCat.Fun) (F.map (eqToHom ⋯)) :
        (a : (fun (X : Type v) ↦ X) (F.obj (op (CostructuredArrow.mk q.fst)))) →
          (fun (X : Type v) ↦ X) (F.obj (op (CostructuredArrow.mk p.fst))))
      q.snd =
    p.snd
⊢ p = q
have rcases : ∀ (p : A.obj (op X))
  (p' : F.obj (op (CostructuredArrow.mk (Y := X) ((Equiv.symm yonedaEquiv : (a : A.obj (op X)) → yoneda.obj X ⟶ A) p))))
  (h :
    fst (F := F)
        (Sigma.mk (β := fun (s : A.obj (op X)) ↦
          F.obj
            (op (CostructuredArrow.mk (Y := X) ((Equiv.symm yonedaEquiv : (a : A.obj (op X)) → yoneda.obj X ⟶ A) s))))
          p p') =
      q.fst)
  (h' :
    Eq (α :=
      (fun (X : Type v) ↦ X)
        (F.obj
          (op
            (CostructuredArrow.mk
              (fst (F := F)
                (Sigma.mk (β := fun (s : A.obj (op X)) ↦
                  F.obj
                    (op
                      (CostructuredArrow.mk (Y := X)
                        ((Equiv.symm yonedaEquiv : (a : A.obj (op X)) → yoneda.obj X ⟶ A) s))))
                  p p'))))))
      ((ConcreteCategory.hom (FC := TypeCat.Fun) (F.map (eqToHom sorry)) :
            (a : (fun (X : Type v) ↦ X) (F.obj (op (CostructuredArrow.mk q.fst)))) →
              (fun (X : Type v) ↦ X) (F.obj (op (CostructuredArrow.mk (fst ⟨p, p'⟩)))))
          q.snd :
        (fun (X : Type v) ↦ X)
          (F.obj
            (op
              (CostructuredArrow.mk
                (fst (F := F)
                  (Sigma.mk (β := fun (s : A.obj (op X)) ↦
                    F.obj
                      (op
                        (CostructuredArrow.mk (Y := X)
                          ((Equiv.symm yonedaEquiv : (a : A.obj (op X)) → yoneda.obj X ⟶ A) s))))
                    p p'))))))
      (snd
          (Sigma.mk (β := fun (s : A.obj (op X)) ↦
            F.obj
              (op (CostructuredArrow.mk (Y := X) ((Equiv.symm yonedaEquiv : (a : A.obj (op X)) → yoneda.obj X ⟶ A) s))))
            p p') :
        F.obj
          (op
            (CostructuredArrow.mk
              (fst (F := F)
                (Sigma.mk (β := fun (s : A.obj (op X)) ↦
                  F.obj
                    (op
                      (CostructuredArrow.mk (Y := X)
                        ((Equiv.symm yonedaEquiv : (a : A.obj (op X)) → yoneda.obj X ⟶ A) s))))
                  p p')))))),
  Sigma.mk (β := fun (s : A.obj (op X)) ↦
      F.obj (op (CostructuredArrow.mk (Y := X) ((Equiv.symm yonedaEquiv : (a : A.obj (op X)) → yoneda.obj X ⟶ A) s)))) p
      p' =
    q
Tactic: rcases p with ⟨p, p'⟩
Cleared hypotheses: p, h, h'
-- Lean 4 proof step from Mathlib.CategoryTheory.Comma.Presheaf.Basic (goal)
Goal:
C : Type u
inst : Category.{v, u} C
A : Functor.{v, v, u, v + 1} Cᵒᵖ (Type v)
F : (CostructuredArrow yoneda A)ᵒᵖ ⥤ Type v
X : C
G : (CostructuredArrow yoneda A)ᵒᵖ ⥤ Type v
η : F ⟶ G
p : YonedaCollection F X
⊢ op (CostructuredArrow.mk p.fst) = op (CostructuredArrow.mk (map₁ η p).fst)
have rw : op (CostructuredArrow.mk p.fst) = op (CostructuredArrow.mk p.fst)
Tactic: rw [YonedaCollection.map₁_fst]
-- Lean 4 proof step from Mathlib.CategoryTheory.Comma.Presheaf.Basic (goal)
Goal:
C : Type u
inst : Category.{v, u} C
A : Functor.{v, v, u, v + 1} Cᵒᵖ (Type v)
F : (CostructuredArrow yoneda A)ᵒᵖ ⥤ Type v
X Y : C
f : X ⟶ Y
p : YonedaCollection F Y
⊢ op (CostructuredArrow.mk (Functor.map yoneda f ≫ p.fst)) = op (CostructuredArrow.mk (map₂ F f p).fst)
have rw : op (CostructuredArrow.mk (Functor.map yoneda f ≫ p.fst)) = op (CostructuredArrow.mk (Functor.map yoneda f ≫ p.fst))
Tactic: rw [YonedaCollection.map₂_fst f]
-- Lean 4 proof step from Mathlib.CategoryTheory.Comma.Presheaf.Basic (goal)
Goal:
C : Type u
inst : Category.{v, u} C
A : Functor.{v, v, u, v + 1} Cᵒᵖ (Type v)
F : (CostructuredArrow yoneda A)ᵒᵖ ⥤ Type v
X : C
G H : (CostructuredArrow yoneda A)ᵒᵖ ⥤ Type v
η : F ⟶ G
μ : G ⟶ H
⊢ Eq (α := YonedaCollection F X → YonedaCollection H X) (map₁ (η ≫ μ)) (map₁ μ ∘ map₁ η)
have h : ∀ (x : YonedaCollection F X) (x_1 : Cᵒᵖ) (x_2 : Functor.obj (Functor.obj yoneda X) x_1),
  (ConcreteCategory.hom (FC := TypeCat.Fun) ((map₁ (η ≫ μ) x).fst.app x_1)).toFun x_2 =
    (ConcreteCategory.hom (FC := TypeCat.Fun) (((map₁ μ ∘ map₁ η) x).fst.app x_1)).toFun x_2
Tactic: ext
-- Lean 4 proof step from Mathlib.CategoryTheory.Comma.Presheaf.Basic (goal)
Goal:
C : Type u
inst : Category.{v, u} C
A : Functor.{v, v, u, v + 1} Cᵒᵖ (Type v)
F : (CostructuredArrow yoneda A)ᵒᵖ ⥤ Type v
X : C
G H : (CostructuredArrow yoneda A)ᵒᵖ ⥤ Type v
η : F ⟶ G
μ : G ⟶ H
h :
  ∀ (x : YonedaCollection F X) (x_1 : Cᵒᵖ) (x_2 : Functor.obj (Functor.obj yoneda X) x_1),
    (ConcreteCategory.hom (FC := TypeCat.Fun) ((map₁ (η ≫ μ) x).fst.app x_1)).toFun x_2 =
      (ConcreteCategory.hom (FC := TypeCat.Fun) (((map₁ μ ∘ map₁ η) x).fst.app x_1)).toFun x_2
⊢ Eq (α := YonedaCollection F X → YonedaCollection H X) (map₁ (η ≫ μ)) (map₁ μ ∘ map₁ η)
have h₁ : ∀ (x : YonedaCollection F X),
  (ConcreteCategory.hom (FC := TypeCat.Fun) (H.map (eqToHom sorry)) :
        (a : (fun (X : Type v) ↦ X) (H.obj (op (CostructuredArrow.mk ((map₁ μ ∘ map₁ η) x).fst)))) →
          (fun (X : Type v) ↦ X) (H.obj (op (CostructuredArrow.mk (map₁ (η ≫ μ) x).fst))))
      ((map₁ μ ∘ map₁ η) x).snd =
    (map₁ (η ≫ μ) x).snd
Tactic: ext
-- Lean 4 proof step from Mathlib.CategoryTheory.Comma.Presheaf.Basic (goal)
Goal:
C : Type u
inst : Category.{v, u} C
A : Functor.{v, v, u, v + 1} Cᵒᵖ (Type v)
F : (CostructuredArrow yoneda A)ᵒᵖ ⥤ Type v
X : C
⊢ map₂ F (𝟙 X) = id
have h : ∀ (x : YonedaCollection F X) (x_1 : Cᵒᵖ) (x_2 : Functor.obj (Functor.obj yoneda X) x_1),
  (ConcreteCategory.hom (FC := TypeCat.Fun) ((map₂ F (𝟙 X) x).fst.app x_1)).toFun x_2 =
    (ConcreteCategory.hom (FC := TypeCat.Fun) ((id x).fst.app x_1)).toFun x_2
Tactic: ext
-- Lean 4 proof step from Mathlib.CategoryTheory.Comma.Presheaf.Basic (goal)
Goal:
C : Type u
inst : Category.{v, u} C
A : Functor.{v, v, u, v + 1} Cᵒᵖ (Type v)
F : (CostructuredArrow yoneda A)ᵒᵖ ⥤ Type v
X : C
h :
  ∀ (x : YonedaCollection F X) (x_1 : Cᵒᵖ) (x_2 : Functor.obj (Functor.obj yoneda X) x_1),
    (ConcreteCategory.hom (FC := TypeCat.Fun) ((map₂ F (𝟙 X) x).fst.app x_1)).toFun x_2 =
      (ConcreteCategory.hom (FC := TypeCat.Fun) ((id x).fst.app x_1)).toFun x_2
⊢ map₂ F (𝟙 X) = id
have h₁ : ∀ (x : YonedaCollection F X),
  (ConcreteCategory.hom (FC := TypeCat.Fun) (F.map (eqToHom sorry)) :
        (a : (fun (X : Type v) ↦ X) (F.obj (op (CostructuredArrow.mk (id x).fst)))) →
          (fun (X : Type v) ↦ X) (F.obj (op (CostructuredArrow.mk (map₂ F (𝟙 X) x).fst))))
      (id x).snd =
    (map₂ F (𝟙 X) x).snd
Tactic: ext
-- Lean 4 proof step from Mathlib.CategoryTheory.Comma.Presheaf.Basic (goal)
Goal:
C : Type u
inst : Category.{v, u} C
A : Functor.{v, v, u, v + 1} Cᵒᵖ (Type v)
F : (CostructuredArrow yoneda A)ᵒᵖ ⥤ Type v
X Y Z : C
f : X ⟶ Y
g : Y ⟶ Z
⊢ map₂ F (f ≫ g) = map₂ F f ∘ map₂ F g
have h : ∀ (x : YonedaCollection F Z) (x_1 : Cᵒᵖ) (x_2 : Functor.obj (Functor.obj yoneda X) x_1),
  (ConcreteCategory.hom (FC := TypeCat.Fun) ((map₂ F (f ≫ g) x).fst.app x_1)).toFun x_2 =
    (ConcreteCategory.hom (FC := TypeCat.Fun) (((map₂ F f ∘ map₂ F g) x).fst.app x_1)).toFun x_2
Tactic: ext
-- Lean 4 proof step from Mathlib.CategoryTheory.Comma.Presheaf.Basic (goal)
Goal:
C : Type u
inst : Category.{v, u} C
A : Functor.{v, v, u, v + 1} Cᵒᵖ (Type v)
F : (CostructuredArrow yoneda A)ᵒᵖ ⥤ Type v
X Y Z : C
f : X ⟶ Y
g : Y ⟶ Z
h :
  ∀ (x : YonedaCollection F Z) (x_1 : Cᵒᵖ) (x_2 : Functor.obj (Functor.obj yoneda X) x_1),
    (ConcreteCategory.hom (FC := TypeCat.Fun) ((map₂ F (f ≫ g) x).fst.app x_1)).toFun x_2 =
      (ConcreteCategory.hom (FC := TypeCat.Fun) (((map₂ F f ∘ map₂ F g) x).fst.app x_1)).toFun x_2
⊢ map₂ F (f ≫ g) = map₂ F f ∘ map₂ F g
have h₁ : ∀ (x : YonedaCollection F Z),
  (ConcreteCategory.hom (FC := TypeCat.Fun) (F.map (eqToHom sorry)) :
        (a : (fun (X : Type v) ↦ X) (F.obj (op (CostructuredArrow.mk ((map₂ F f ∘ map₂ F g) x).fst)))) →
          (fun (X : Type v) ↦ X) (F.obj (op (CostructuredArrow.mk (map₂ F (f ≫ g) x).fst))))
      ((map₂ F f ∘ map₂ F g) x).snd =
    (map₂ F (f ≫ g) x).snd
Tactic: ext
-- Lean 4 proof step from Mathlib.CategoryTheory.Comma.Presheaf.Basic (goal)
Goal:
C : Type u
inst : Category.{v, u} C
A : Functor.{v, v, u, v + 1} Cᵒᵖ (Type v)
F : (CostructuredArrow yoneda A)ᵒᵖ ⥤ Type v
X : C
G : (CostructuredArrow yoneda A)ᵒᵖ ⥤ Type v
η : F ⟶ G
Y : C
f : X ⟶ Y
p : YonedaCollection F Y
⊢ map₂ G f (map₁ η p) = map₁ η (map₂ F f p)
have h : ∀ (x : Cᵒᵖ) (x_1 : Functor.obj (Functor.obj yoneda X) x),
  (ConcreteCategory.hom (FC := TypeCat.Fun) ((map₂ G f (map₁ η p)).fst.app x)).toFun x_1 =
    (ConcreteCategory.hom (FC := TypeCat.Fun) ((map₁ η (map₂ F f p)).fst.app x)).toFun x_1
Tactic: ext
-- Lean 4 proof step from Mathlib.CategoryTheory.Comma.Presheaf.Basic (goal)
Goal:
C : Type u
inst : Category.{v, u} C
A : Functor.{v, v, u, v + 1} Cᵒᵖ (Type v)
F : (CostructuredArrow yoneda A)ᵒᵖ ⥤ Type v
X : C
G : (CostructuredArrow yoneda A)ᵒᵖ ⥤ Type v
η : F ⟶ G
Y : C
f : X ⟶ Y
p : YonedaCollection F Y
h :
  ∀ (x : Cᵒᵖ) (x_1 : Functor.obj (Functor.obj yoneda X) x),
    (ConcreteCategory.hom (FC := TypeCat.Fun) ((map₂ G f (map₁ η p)).fst.app x)).toFun x_1 =
      (ConcreteCategory.hom (FC := TypeCat.Fun) ((map₁ η (map₂ F f p)).fst.app x)).toFun x_1
⊢ map₂ G f (map₁ η p) = map₁ η (map₂ F f p)
have h' : (ConcreteCategory.hom (FC := TypeCat.Fun) (G.map (eqToHom sorry)) :
      (a : (fun (X : Type v) ↦ X) (G.obj (op (CostructuredArrow.mk (map₁ η (map₂ F f p)).fst)))) →
        (fun (X : Type v) ↦ X) (G.obj (op (CostructuredArrow.mk (map₂ G f (map₁ η p)).fst))))
    (map₁ η (map₂ F f p)).snd =
  (map₂ G f (map₁ η p)).snd
Tactic: ext
-- Lean 4 proof step from Mathlib.CategoryTheory.Comma.Presheaf.Basic (goal)
Goal:
C : Type u
inst : Category.{v, u} C
A : Functor.{v, v, u, v + 1} Cᵒᵖ (Type v)
F G : (CostructuredArrow yoneda A)ᵒᵖ ⥤ Type v
η : F ⟶ G
X Y : Cᵒᵖ
f : X ⟶ Y
⊢ (yonedaCollectionPresheaf A F).map f ≫ ↾YonedaCollection.map₁ η =
    ↾YonedaCollection.map₁ η ≫ (yonedaCollectionPresheaf A G).map f
have h : ∀ (x : (yonedaCollectionPresheaf A F).obj X),
  (ConcreteCategory.hom (FC := TypeCat.Fun) ((yonedaCollectionPresheaf A F).map f ≫ ↾YonedaCollection.map₁ η)).toFun x =
    (ConcreteCategory.hom (FC := TypeCat.Fun) (↾YonedaCollection.map₁ η ≫ (yonedaCollectionPresheaf A G).map f)).toFun x
Tactic: ext
-- Lean 4 proof step from Mathlib.CategoryTheory.Comma.Presheaf.Basic (goal)
Goal:
C : Type u
inst : Category.{v, u} C
A F : Functor.{v, v, u, v + 1} Cᵒᵖ (Type v)
η : F ⟶ A
X : C
⊢ unitBackward η X ∘ unitForward η X = id
have refine_1 : ∀ (p : YonedaCollection (restrictedYonedaObj η) X), ((unitBackward η X ∘ unitForward η X) p).fst = (id p).fst
Tactic: refine funext fun p => YonedaCollection.ext ?_ (OverArrows.ext ?_)
-- Lean 4 proof step from Mathlib.CategoryTheory.Comma.Presheaf.Basic (goal)
Goal:
C : Type u
inst : Category.{v, u} C
A F : Functor.{v, v, u, v + 1} Cᵒᵖ (Type v)
η : F ⟶ A
X : C
refine_1 : ∀ (p : YonedaCollection (restrictedYonedaObj η) X), ((unitBackward η X ∘ unitForward η X) p).fst = (id p).fst
⊢ unitBackward η X ∘ unitForward η X = id
have refine_2 : ∀ (p : YonedaCollection (restrictedYonedaObj η) X),
  OverArrows.val (A := A) (η := η) (s :=
      (unop (op (CostructuredArrow.mk ((unitBackward η X ∘ unitForward η X) p).fst))).hom)
      ((ConcreteCategory.hom (FC := TypeCat.Fun) ((restrictedYonedaObj η).map (eqToHom sorry)) :
          (a : (fun (X : Type v) ↦ X) ((restrictedYonedaObj η).obj (op (CostructuredArrow.mk (id p).fst)))) →
            (fun (X : Type v) ↦ X)
              ((restrictedYonedaObj η).obj (op (CostructuredArrow.mk ((unitBackward η X ∘ unitForward η X) p).fst))))
        (id p).snd) =
    OverArrows.val ((unitBackward η X ∘ unitForward η X) p).snd
Tactic: refine funext fun p => YonedaCollection.ext ?_ (OverArrows.ext ?_)
-- Lean 4 proof step from Mathlib.CategoryTheory.Comma.Presheaf.Basic (goal)
Goal:
C : Type u
inst : Category.{v, u} C
A : Functor.{v, v, u, v + 1} Cᵒᵖ (Type v)
F : (CostructuredArrow yoneda A)ᵒᵖ ⥤ Type v
s t : (CostructuredArrow yoneda A)ᵒᵖ
f : t ⟶ s
x : F.obj t
⊢ counitForward F (unop s)
      ((ConcreteCategory.hom (FC := TypeCat.Fun) (F.map f) :
          (a : (fun (X : Type v) ↦ X) (F.obj t)) → (fun (X : Type v) ↦ X) (F.obj s))
        x) =
    (counitForward F (unop t) x).map₂ f.unop.left ⋯
have refine : Eq (α :=
  (fun (X : Type v) ↦ X)
    (F.obj
      (op
        (CostructuredArrow.mk (Y := unop (op (unop s).left))
          (YonedaCollection.fst
            (OverArrows.val
              (counitForward F (unop s)
                ((ConcreteCategory.hom (FC := TypeCat.Fun) (F.map f) :
                    (a : (fun (X : Type v) ↦ X) (F.obj t)) → (fun (X : Type v) ↦ X) (F.obj s))
                  x))))))))
  ((ConcreteCategory.hom (FC := TypeCat.Fun) (F.map (eqToHom sorry)) :
        (a :
            (fun (X : Type v) ↦ X)
              (F.obj
                (op
                  (CostructuredArrow.mk
                    (YonedaCollection.fst ((counitForward F (unop t) x).map₂ f.unop.left sorry).val))))) →
          (fun (X : Type v) ↦ X)
            (F.obj
              (op
                (CostructuredArrow.mk
                  (YonedaCollection.fst
                    (counitForward F (unop s)
                        ((ConcreteCategory.hom (F.map f) :
                            (a : (fun (X : Type v) ↦ X) (F.obj t)) → (fun (X : Type v) ↦ X) (F.obj s))
                          x)).val)))))
      (YonedaCollection.snd ((counitForward F (unop t) x).map₂ f.unop.left sorry).val) :
    (fun (X : Type v) ↦ X)
      (F.obj
        (op
          (CostructuredArrow.mk (Y := unop (op (unop s).left))
            (YonedaCollection.fst
              (OverArrows.val
                (counitForward F (unop s)
                  ((ConcreteCategory.hom (FC := TypeCat.Fun) (F.map f) :
                      (a : (fun (X : Type v) ↦ X) (F.obj t)) → (fun (X : Type v) ↦ X) (F.obj s))
                    x))))))))
  (YonedaCollection.snd
      (OverArrows.val
        (counitForward F (unop s)
          ((ConcreteCategory.hom (FC := TypeCat.Fun) (F.map f) :
              (a : (fun (X : Type v) ↦ X) (F.obj t)) → (fun (X : Type v) ↦ X) (F.obj s))
            x))) :
    F.obj
      (op
        (CostructuredArrow.mk (Y := unop (op (unop s).left))
          (YonedaCollection.fst
            (OverArrows.val
              (counitForward F (unop s)
                ((ConcreteCategory.hom (FC := TypeCat.Fun) (F.map f) :
                    (a : (fun (X : Type v) ↦ X) (F.obj t)) → (fun (X : Type v) ↦ X) (F.obj s))
                  x)))))))
Tactic: refine OverArrows.ext <| YonedaCollection.ext (by simp) ?_
-- Lean 4 proof step from Mathlib.CategoryTheory.Comma.Presheaf.Basic (goal)
Goal:
C : Type u
inst : Category.{v, u} C
A : Functor.{v, v, u, v + 1} Cᵒᵖ (Type v)
F : (CostructuredArrow yoneda A)ᵒᵖ ⥤ Type v
s t : (CostructuredArrow yoneda A)ᵒᵖ
f : t ⟶ s
x : F.obj t
⊢ (CostructuredArrow.mkPrecomp (unop t).hom f.unop.left).op = f ≫ eqToHom ⋯
have a : (CostructuredArrow.mkPrecomp (unop t).hom f.unop.left).op.unop = Quiver.Hom.unop (f ≫ eqToHom sorry)
Tactic: apply Quiver.Hom.unop_inj
-- Lean 4 proof step from Mathlib.CategoryTheory.Comma.Presheaf.Basic (hypothesis)
Goal:
C : Type u
inst : Category.{v, u} C
A : Functor.{v, v, u, v + 1} Cᵒᵖ (Type v)
F : (CostructuredArrow yoneda A)ᵒᵖ ⥤ Type v
s t : (CostructuredArrow yoneda A)ᵒᵖ
f : t ⟶ s
x : F.obj t
⊢ Eq (α :=
    (fun (X : Type v) ↦ X)
      (F.obj
        (op
          (CostructuredArrow.mk (Y := unop (op (unop s).left))
            (YonedaCollection.fst
              (OverArrows.val
                (counitForward F (unop s)
                  ((ConcreteCategory.hom (FC := TypeCat.Fun) (F.map f) :
                      (a : (fun (X : Type v) ↦ X) (F.obj t)) → (fun (X : Type v) ↦ X) (F.obj s))
                    x))))))))
    ((ConcreteCategory.hom (FC := TypeCat.Fun) (F.map (eqToHom ⋯)) :
          (a :
              (fun (X : Type v) ↦ X)
                (F.obj
                  (op
                    (CostructuredArrow.mk
                      (YonedaCollection.fst ((counitForward F (unop t) x).map₂ f.unop.left ⋯).val))))) →
            (fun (X : Type v) ↦ X)
              (F.obj
                (op
                  (CostructuredArrow.mk
                    (YonedaCollection.fst
                      (counitForward F (unop s)
                          ((ConcreteCategory.hom (F.map f) :
                              (a : (fun (X : Type v) ↦ X) (F.obj t)) → (fun (X : Type v) ↦ X) (F.obj s))
                            x)).val)))))
        (YonedaCollection.snd ((counitForward F (unop t) x).map₂ f.unop.left ⋯).val) :
      (fun (X : Type v) ↦ X)
        (F.obj
          (op
            (CostructuredArrow.mk (Y := unop (op (unop s).left))
              (YonedaCollection.fst
                (OverArrows.val
                  (counitForward F (unop s)
                    ((ConcreteCategory.hom (FC := TypeCat.Fun) (F.map f) :
                        (a : (fun (X : Type v) ↦ X) (F.obj t)) → (fun (X : Type v) ↦ X) (F.obj s))
                      x))))))))
    (YonedaCollection.snd
        (OverArrows.val
          (counitForward F (unop s)
            ((ConcreteCategory.hom (FC := TypeCat.Fun) (F.map f) :
                (a : (fun (X : Type v) ↦ X) (F.obj t)) → (fun (X : Type v) ↦ X) (F.obj s))
              x))) :
      F.obj
        (op
          (CostructuredArrow.mk (Y := unop (op (unop s).left))
            (YonedaCollection.fst
              (OverArrows.val
                (counitForward F (unop s)
                  ((ConcreteCategory.hom (FC := TypeCat.Fun) (F.map f) :
                      (a : (fun (X : Type v) ↦ X) (F.obj t)) → (fun (X : Type v) ↦ X) (F.obj s))
                    x)))))))
have this : (CostructuredArrow.mkPrecomp (unop t).hom f.unop.left).op = f ≫ eqToHom sorry
Tactic: have : (CostructuredArrow.mkPrecomp t.unop.hom f.unop.left).op =
      f ≫ eqToHom (by simp [← CostructuredArrow.eq_mk]) := by
    apply Quiver.Hom.unop_inj
    simp
-- Lean 4 proof step from Mathlib.CategoryTheory.Comma.Presheaf.Basic (goal)
Goal:
C : Type u
inst : Category.{v, u} C
A : Functor.{v, v, u, v + 1} Cᵒᵖ (Type v)
F : (CostructuredArrow yoneda A)ᵒᵖ ⥤ Type v
s t : (CostructuredArrow yoneda A)ᵒᵖ
f : t ⟶ s
x : F.obj t
this : (CostructuredArrow.mkPrecomp (unop t).hom f.unop.left).op = f ≫ eqToHom ⋯
⊢ t = op (CostructuredArrow.mk (YonedaCollection.fst (counitForward F (unop t) x).val))
have simp : t = op (CostructuredArrow.mk (unop t).hom)
Tactic: simp