awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.CategoryTheory.Monoidal.Cartesian.Normal (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : CartesianMonoidalCategory.{v_1, u_1} C
G H : C
inst_2 : GrpObj G
inst_3 : GrpObj H
φ : H ⟶ G
inst_4 : IsMonHom φ
inst_5 : Mono φ
hnormal : ∀ (X : C), Subgroup.Normal.{v_1} (G := X ⟶ G) (MonoidHom.range (G := X ⟶ H) (monoidHom φ X))
h' : (X : C) → (g : X ⟶ G) → (h : X ⟶ H) → X ⟶ H
hh' : ∀ (X : C) (g : X ⟶ G) (h : X ⟶ H), h' X g h ≫ φ = g * h ≫ φ * g⁻¹
⊢ Quiver.Hom (V := Functor.{v_1, v_1, u_1, v_1 + 1} Cᵒᵖ (Type v_1))
    (Functor.obj yoneda (tensorObj.{v_1, u_1} (𝒞 := inst) G H)) (Functor.obj yoneda H)
have refine_2 : ∀ (X Y : Cᵒᵖ) (f : X ⟶ Y),
  Eq (α :=
    Functor.obj (Functor.obj yoneda (tensorObj.{v_1, u_1} (𝒞 := inst) G H)) X ⟶ Functor.obj (Functor.obj yoneda H) Y)
    (Functor.map (Functor.obj yoneda (tensorObj.{v_1, u_1} (𝒞 := inst) G H)) f ≫
      (fun (X : Cᵒᵖ) ↦
          (↾fun (x : Functor.obj (Functor.obj yoneda (tensorObj.{v_1, u_1} (𝒞 := inst) G H)) X) ↦
              h' (Opposite.unop X) (x ≫ fst G H) (x ≫ snd G H) :
            Functor.obj (Functor.obj yoneda (tensorObj.{v_1, u_1} (𝒞 := inst) G H)) X ⟶
              Functor.obj (Functor.obj yoneda H) X))
        Y)
    ((fun (X : Cᵒᵖ) ↦
          (↾fun (x : Functor.obj (Functor.obj yoneda (tensorObj.{v_1, u_1} (𝒞 := inst) G H)) X) ↦
              h' (Opposite.unop X) (x ≫ fst G H) (x ≫ snd G H) :
            Functor.obj (Functor.obj yoneda (tensorObj.{v_1, u_1} (𝒞 := inst) G H)) X ⟶
              Functor.obj (Functor.obj yoneda H) X))
        X ≫
      Functor.map (Functor.obj yoneda H) f)
Tactic: refine ⟨fun X ↦ ↾fun x ↦ h' _ (x ≫ fst _ _) (x ≫ snd _ _), fun X Y f ↦ ?_⟩
-- Lean 4 proof step from Mathlib.CategoryTheory.Monoidal.Cartesian.Normal (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : CartesianMonoidalCategory.{v_1, u_1} C
G H : C
inst_2 : GrpObj G
inst_3 : GrpObj H
φ : H ⟶ G
inst_4 : IsMonHom φ
inst_5 : Mono φ
hnormal : ∀ (X : C), Subgroup.Normal.{v_1} (G := X ⟶ G) (MonoidHom.range (G := X ⟶ H) (monoidHom φ X))
h' : (X : C) → (g : X ⟶ G) → (h : X ⟶ H) → X ⟶ H
hh' : ∀ (X : C) (g : X ⟶ G) (h : X ⟶ H), h' X g h ≫ φ = g * h ≫ φ * g⁻¹
X Y : Cᵒᵖ
f : X ⟶ Y
⊢ Eq (α :=
    Functor.obj (Functor.obj yoneda (tensorObj.{v_1, u_1} (𝒞 := inst) G H)) X ⟶ Functor.obj (Functor.obj yoneda H) Y)
    (Functor.map (Functor.obj yoneda (tensorObj.{v_1, u_1} (𝒞 := inst) G H)) f ≫
      (fun (X : Cᵒᵖ) ↦
          (↾fun (x : Functor.obj (Functor.obj yoneda (tensorObj.{v_1, u_1} (𝒞 := inst) G H)) X) ↦
              h' (Opposite.unop X) (x ≫ fst G H) (x ≫ snd G H) :
            Functor.obj (Functor.obj yoneda (tensorObj.{v_1, u_1} (𝒞 := inst) G H)) X ⟶
              Functor.obj (Functor.obj yoneda H) X))
        Y)
    ((fun (X : Cᵒᵖ) ↦
          (↾fun (x : Functor.obj (Functor.obj yoneda (tensorObj.{v_1, u_1} (𝒞 := inst) G H)) X) ↦
              h' (Opposite.unop X) (x ≫ fst G H) (x ≫ snd G H) :
            Functor.obj (Functor.obj yoneda (tensorObj.{v_1, u_1} (𝒞 := inst) G H)) X ⟶
              Functor.obj (Functor.obj yoneda H) X))
        X ≫
      Functor.map (Functor.obj yoneda H) f)
have refine_2 : ∀ (x : Functor.obj (Functor.obj yoneda (tensorObj.{v_1, u_1} (𝒞 := inst) G H)) X),
  Eq (α := Functor.obj (Functor.obj yoneda H) Y)
    (TypeCat.Fun.toFun
      (ConcreteCategory.hom (FC := TypeCat.Fun)
        (Functor.map (Functor.obj yoneda (tensorObj.{v_1, u_1} (𝒞 := inst) G H)) f ≫
          (fun (X : Cᵒᵖ) ↦
              (↾fun (x : Functor.obj (Functor.obj yoneda (tensorObj.{v_1, u_1} (𝒞 := inst) G H)) X) ↦
                  h' (Opposite.unop X) (x ≫ fst G H) (x ≫ snd G H) :
                Functor.obj (Functor.obj yoneda (tensorObj.{v_1, u_1} (𝒞 := inst) G H)) X ⟶
                  Functor.obj (Functor.obj yoneda H) X))
            Y))
      x)
    (TypeCat.Fun.toFun
      (ConcreteCategory.hom (FC := TypeCat.Fun)
        ((fun (X : Cᵒᵖ) ↦
              (↾fun (x : Functor.obj (Functor.obj yoneda (tensorObj.{v_1, u_1} (𝒞 := inst) G H)) X) ↦
                  h' (Opposite.unop X) (x ≫ fst G H) (x ≫ snd G H) :
                Functor.obj (Functor.obj yoneda (tensorObj.{v_1, u_1} (𝒞 := inst) G H)) X ⟶
                  Functor.obj (Functor.obj yoneda H) X))
            X ≫
          Functor.map (Functor.obj yoneda H) f))
      x)
Tactic: ext
-- Lean 4 proof step from Mathlib.CategoryTheory.Monoidal.Cartesian.Normal (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : CartesianMonoidalCategory.{v_1, u_1} C
G H : C
inst_2 : GrpObj G
inst_3 : GrpObj H
φ : H ⟶ G
inst_4 : IsMonHom φ
inst_5 : Mono φ
hnormal : ∀ (X : C), Subgroup.Normal.{v_1} (G := X ⟶ G) (MonoidHom.range (G := X ⟶ H) (monoidHom φ X))
h' : (X : C) → (g : X ⟶ G) → (h : X ⟶ H) → X ⟶ H
hh' : ∀ (X : C) (g : X ⟶ G) (h : X ⟶ H), h' X g h ≫ φ = g * h ≫ φ * g⁻¹
⊢ Eq (α := tensorObj.{v_1, u_1} (𝒞 := inst) G H ⟶ G)
    (Functor.FullyFaithful.preimage Yoneda.fullyFaithful
        {
          app := fun (X : Cᵒᵖ) ↦
            ↾fun (x : Functor.obj (Functor.obj yoneda (tensorObj.{v_1, u_1} (𝒞 := inst) G H)) X) ↦
              h' (Opposite.unop X) (x ≫ fst G H) (x ≫ snd G H),
          naturality := ⋯ } ≫
      φ)
    (G ◁ φ ≫ conj G)
have refine_2 : Eq (α :=
  Quiver.Hom (V := Functor.{v_1, v_1, u_1, v_1 + 1} Cᵒᵖ (Type v_1))
    (Functor.obj yoneda (tensorObj.{v_1, u_1} (𝒞 := inst) G H)) (Functor.obj yoneda G))
  (Functor.map yoneda
    (Functor.FullyFaithful.preimage Yoneda.fullyFaithful
        {
          app := fun (X : Cᵒᵖ) ↦
            ↾fun (x : Functor.obj (Functor.obj yoneda (tensorObj.{v_1, u_1} (𝒞 := inst) G H)) X) ↦
              h' (Opposite.unop X) (x ≫ fst G H) (x ≫ snd G H),
          naturality := sorry } ≫
      φ))
  (Functor.map yoneda (G ◁ φ ≫ conj G))
Tactic: refine yoneda.map_injective ?_
-- Lean 4 proof step from Mathlib.CategoryTheory.Monoidal.Cartesian.Normal (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : CartesianMonoidalCategory.{v_1, u_1} C
G H : C
inst_2 : GrpObj G
inst_3 : GrpObj H
φ : H ⟶ G
inst_4 : IsMonHom φ
inst_5 : Mono φ
hnormal : ∀ (X : C), Subgroup.Normal.{v_1} (G := X ⟶ G) (MonoidHom.range (G := X ⟶ H) (monoidHom φ X))
h' : (X : C) → (g : X ⟶ G) → (h : X ⟶ H) → X ⟶ H
hh' : ∀ (X : C) (g : X ⟶ G) (h : X ⟶ H), h' X g h ≫ φ = g * h ≫ φ * g⁻¹
⊢ Eq (α :=
    Quiver.Hom (V := Functor.{v_1, v_1, u_1, v_1 + 1} Cᵒᵖ (Type v_1))
      (Functor.obj yoneda (tensorObj.{v_1, u_1} (𝒞 := inst) G H)) (Functor.obj yoneda G))
    (Functor.map yoneda
      (Functor.FullyFaithful.preimage Yoneda.fullyFaithful
          {
            app := fun (X : Cᵒᵖ) ↦
              ↾fun (x : Functor.obj (Functor.obj yoneda (tensorObj.{v_1, u_1} (𝒞 := inst) G H)) X) ↦
                h' (Opposite.unop X) (x ≫ fst G H) (x ≫ snd G H),
            naturality := ⋯ } ≫
        φ))
    (Functor.map yoneda (G ◁ φ ≫ conj G))
have refine_2 : ∀ (x : Cᵒᵖ) (x_1 : Functor.obj (Functor.obj yoneda (tensorObj.{v_1, u_1} (𝒞 := inst) G H)) x),
  TypeCat.Fun.toFun
      (ConcreteCategory.hom (FC := TypeCat.Fun)
        (NatTrans.app
          (Functor.map yoneda
            (Functor.FullyFaithful.preimage Yoneda.fullyFaithful
                {
                  app := fun (X : Cᵒᵖ) ↦
                    ↾fun (x : Functor.obj (Functor.obj yoneda (tensorObj.{v_1, u_1} (𝒞 := inst) G H)) X) ↦
                      h' (Opposite.unop X) (x ≫ fst G H) (x ≫ snd G H),
                  naturality := sorry :
                  NatTrans.{v_1, v_1, u_1, v_1 + 1} (Functor.obj yoneda (tensorObj.{v_1, u_1} (𝒞 := inst) G H))
                    (Functor.obj yoneda H) } ≫
              φ))
          x))
      x_1 =
    TypeCat.Fun.toFun (ConcreteCategory.hom (FC := TypeCat.Fun) (NatTrans.app (Functor.map yoneda (G ◁ φ ≫ conj G)) x))
      x_1
Tactic: ext
-- Lean 4 proof step from Mathlib.CategoryTheory.Monoidal.Cartesian.Normal (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : CartesianMonoidalCategory.{v_1, u_1} C
G H : C
inst_2 : GrpObj G
inst_3 : GrpObj H
φ : H ⟶ G
inst_4 : IsMonHom φ
P : C
p : G ⟶ P
inst_5 : GrpObj P
inst_6 : IsMonHom p
h : IsPullback (Y := tensorUnit.{v_1, u_1} C (𝒞 := inst)) φ (toUnit H) p η
⊢ ∃ (ψ : tensorObj.{v_1, u_1} (𝒞 := inst) G H ⟶ H), ψ ≫ φ = G ◁ φ ≫ conj G
have refine_1 : (G ◁ φ ≫ conj G) ≫ p =
  CategoryStruct.comp (Y := tensorUnit.{v_1, u_1} C (𝒞 := inst)) (toUnit (tensorObj.{v_1, u_1} (𝒞 := inst) G H)) η
Tactic: refine ⟨h.lift (G ◁ φ ≫ conj G) (toUnit _) ?_, ?_⟩
-- Lean 4 proof step from Mathlib.CategoryTheory.Monoidal.Cartesian.Normal (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : CartesianMonoidalCategory.{v_1, u_1} C
G H : C
inst_2 : GrpObj G
inst_3 : GrpObj H
φ : H ⟶ G
inst_4 : IsMonHom φ
P : C
p : G ⟶ P
inst_5 : GrpObj P
inst_6 : IsMonHom p
h : IsPullback (Y := tensorUnit.{v_1, u_1} C (𝒞 := inst)) φ (toUnit H) p η
refine_1 :
  (G ◁ φ ≫ conj G) ≫ p =
    CategoryStruct.comp (Y := tensorUnit.{v_1, u_1} C (𝒞 := inst)) (toUnit (tensorObj.{v_1, u_1} (𝒞 := inst) G H)) η
⊢ ∃ (ψ : tensorObj.{v_1, u_1} (𝒞 := inst) G H ⟶ H), ψ ≫ φ = G ◁ φ ≫ conj G
have refine_2 : h.lift (G ◁ φ ≫ conj G) (toUnit (tensorObj.{v_1, u_1} (𝒞 := inst) G H)) sorry ≫ φ = G ◁ φ ≫ conj G
Tactic: refine ⟨h.lift (G ◁ φ ≫ conj G) (toUnit _) ?_, ?_⟩
-- Lean 4 proof step from Mathlib.CategoryTheory.Monoidal.Cartesian.Over (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : HasPullbacks.{v_1, u_1} C
X R : C
f : R ⟶ X
⊢ Eq (α :=
    left.{v_1, u_1} (X := R) ((𝟙_ (Over.{v_1, u_1} R)) (𝒞 := instCategoryOver)) ⟶
      ((pullback f).obj ((𝟙_ (Over.{v_1, u_1} X)) (𝒞 := instCategoryOver))).left)
    (Hom.left (LaxMonoidal.ε (pullback f))) (inv (pullback.snd (𝟙 X) f))
have hom_inv_id : pullback.snd (𝟙 X) f ≫ Hom.left (LaxMonoidal.ε (pullback f)) =
  𝟙 ((pullback f).obj ((𝟙_ (Over.{v_1, u_1} X)) (𝒞 := instCategoryOver))).left
Tactic: apply IsIso.eq_inv_of_hom_inv_id
-- Lean 4 proof step from Mathlib.CategoryTheory.Monoidal.Cartesian.Over (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : HasPullbacks.{v_1, u_1} C
X R : C
f : R ⟶ X
⊢ pullback.snd (𝟙 X) f ≫ Hom.left (LaxMonoidal.ε (pullback f)) =
    𝟙 ((pullback f).obj ((𝟙_ (Over.{v_1, u_1} X)) (𝒞 := instCategoryOver))).left
have hom_inv_id : CategoryStruct.comp (Y := left.{v_1, u_1} (X := R) ((𝟙_ (Over.{v_1, u_1} R)) (𝒞 := instCategoryOver)))
    (Hom.left (OplaxMonoidal.η (pullback f))) (Hom.left (LaxMonoidal.ε (pullback f))) =
  𝟙 ((pullback f).obj ((𝟙_ (Over.{v_1, u_1} X)) (𝒞 := instCategoryOver))).left
Tactic: rw [← η_pullback_left, ← Over.comp_left, Monoidal.η_ε, Over.id_left]
-- Lean 4 proof step from Mathlib.CategoryTheory.Monoidal.Cartesian.Over (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : HasPullbacks.{v_1, u_1} C
X R : C
f : R ⟶ X
hom_inv_id :
  CategoryStruct.comp (Y := left.{v_1, u_1} (X := R) ((𝟙_ (Over.{v_1, u_1} R)) (𝒞 := instCategoryOver)))
      (Hom.left (OplaxMonoidal.η (pullback f))) (Hom.left (LaxMonoidal.ε (pullback f))) =
    𝟙 ((pullback f).obj ((𝟙_ (Over.{v_1, u_1} X)) (𝒞 := instCategoryOver))).left
⊢ pullback.snd (𝟙 X) f ≫ Hom.left (LaxMonoidal.ε (pullback f)) =
    𝟙 ((pullback f).obj ((𝟙_ (Over.{v_1, u_1} X)) (𝒞 := instCategoryOver))).left
have hom_inv_id₁ : Hom.left
    (CategoryStruct.comp (Y := (𝟙_ (Over.{v_1, u_1} R)) (𝒞 := instCategoryOver)) (OplaxMonoidal.η (pullback f))
      (LaxMonoidal.ε (pullback f))) =
  𝟙 ((pullback f).obj ((𝟙_ (Over.{v_1, u_1} X)) (𝒞 := instCategoryOver))).left
Tactic: rw [← η_pullback_left, ← Over.comp_left, Monoidal.η_ε, Over.id_left]
-- Lean 4 proof step from Mathlib.CategoryTheory.Monoidal.Cartesian.Over (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : HasPullbacks.{v_1, u_1} C
X R : C
f : R ⟶ X
hom_inv_id :
  CategoryStruct.comp (Y := left.{v_1, u_1} (X := R) ((𝟙_ (Over.{v_1, u_1} R)) (𝒞 := instCategoryOver)))
      (Hom.left (OplaxMonoidal.η (pullback f))) (Hom.left (LaxMonoidal.ε (pullback f))) =
    𝟙 ((pullback f).obj ((𝟙_ (Over.{v_1, u_1} X)) (𝒞 := instCategoryOver))).left
hom_inv_id₁ :
  Hom.left
      (CategoryStruct.comp (Y := (𝟙_ (Over.{v_1, u_1} R)) (𝒞 := instCategoryOver)) (OplaxMonoidal.η (pullback f))
        (LaxMonoidal.ε (pullback f))) =
    𝟙 ((pullback f).obj ((𝟙_ (Over.{v_1, u_1} X)) (𝒞 := instCategoryOver))).left
⊢ pullback.snd (𝟙 X) f ≫ Hom.left (LaxMonoidal.ε (pullback f)) =
    𝟙 ((pullback f).obj ((𝟙_ (Over.{v_1, u_1} X)) (𝒞 := instCategoryOver))).left
have hom_inv_id₂ : Hom.left (𝟙 ((pullback f).obj ((𝟙_ (Over.{v_1, u_1} X)) (𝒞 := instCategoryOver)))) =
  𝟙 ((pullback f).obj ((𝟙_ (Over.{v_1, u_1} X)) (𝒞 := instCategoryOver))).left
Tactic: rw [← η_pullback_left, ← Over.comp_left, Monoidal.η_ε, Over.id_left]
-- Lean 4 proof step from Mathlib.CategoryTheory.Monoidal.Cartesian.Over (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : HasPullbacks.{v_1, u_1} C
X R : C
f : R ⟶ X
hom_inv_id :
  CategoryStruct.comp (Y := left.{v_1, u_1} (X := R) ((𝟙_ (Over.{v_1, u_1} R)) (𝒞 := instCategoryOver)))
      (Hom.left (OplaxMonoidal.η (pullback f))) (Hom.left (LaxMonoidal.ε (pullback f))) =
    𝟙 ((pullback f).obj ((𝟙_ (Over.{v_1, u_1} X)) (𝒞 := instCategoryOver))).left
hom_inv_id₁ :
  Hom.left
      (CategoryStruct.comp (Y := (𝟙_ (Over.{v_1, u_1} R)) (𝒞 := instCategoryOver)) (OplaxMonoidal.η (pullback f))
        (LaxMonoidal.ε (pullback f))) =
    𝟙 ((pullback f).obj ((𝟙_ (Over.{v_1, u_1} X)) (𝒞 := instCategoryOver))).left
hom_inv_id₂ :
  Hom.left (𝟙 ((pullback f).obj ((𝟙_ (Over.{v_1, u_1} X)) (𝒞 := instCategoryOver)))) =
    𝟙 ((pullback f).obj ((𝟙_ (Over.{v_1, u_1} X)) (𝒞 := instCategoryOver))).left
⊢ pullback.snd (𝟙 X) f ≫ Hom.left (LaxMonoidal.ε (pullback f)) =
    𝟙 ((pullback f).obj ((𝟙_ (Over.{v_1, u_1} X)) (𝒞 := instCategoryOver))).left
have hom_inv_id₃ : Eq.{v_1 + 1} (α :=
  ((pullback f).obj ((𝟙_ (Over.{v_1, u_1} X)) (𝒞 := instCategoryOver))).left ⟶
    ((pullback f).obj ((𝟙_ (Over.{v_1, u_1} X)) (𝒞 := instCategoryOver))).left)
  (𝟙 ((pullback f).obj ((𝟙_ (Over.{v_1, u_1} X)) (𝒞 := instCategoryOver))).left)
  (𝟙 ((pullback f).obj ((𝟙_ (Over.{v_1, u_1} X)) (𝒞 := instCategoryOver))).left)
Tactic: rw [← η_pullback_left, ← Over.comp_left, Monoidal.η_ε, Over.id_left]
-- Lean 4 proof step from Mathlib.CategoryTheory.Monoidal.Cartesian.Over (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : HasPullbacks.{v_1, u_1} C
X R_1 : C
f : R_1 ⟶ X
R S : Over.{v_1, u_1} X
⊢ Eq (α := (tensorObj (𝒞 := instCategoryOver) ((pullback f).obj R) ((pullback f).obj S)).left ⟶ R.left)
    (Hom.left (LaxMonoidal.μ (pullback f) R S) ≫
      pullback.fst (tensorObj (𝒞 := instCategoryOver) R S).hom f ≫ pullback.fst R.hom S.hom)
    (pullback.fst ((pullback f).obj R).hom ((pullback f).obj S).hom ≫ pullback.fst R.hom f)
have rw : Hom.left (𝟙 ((pullback f).obj (tensorObj (𝒞 := instCategoryOver) R S))) ≫
    pullback.fst (tensorObj (𝒞 := instCategoryOver) R S).hom f ≫ pullback.fst R.hom S.hom =
  CategoryStruct.comp (Y := (tensorObj (𝒞 := instCategoryOver) ((pullback f).obj R) ((pullback f).obj S)).left)
    (Hom.left (Iso.hom (prodComparisonIso (pullback f) R S)))
    (pullback.fst ((pullback f).obj R).hom ((pullback f).obj S).hom ≫ pullback.fst R.hom f)
Tactic: rw [Monoidal.μ_of_cartesianMonoidalCategory,
    ← cancel_epi (prodComparisonIso (Over.pullback f) R S).hom.left, ← Over.comp_left_assoc,
    Iso.hom_inv_id]
-- Lean 4 proof step from Mathlib.CategoryTheory.Monoidal.Cartesian.Over (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : HasPullbacks.{v_1, u_1} C
X R_1 : C
f : R_1 ⟶ X
R S : Over.{v_1, u_1} X
⊢ Eq (α := (tensorObj (𝒞 := instCategoryOver) ((pullback f).obj R) ((pullback f).obj S)).left ⟶ S.left)
    (Hom.left (LaxMonoidal.μ (pullback f) R S) ≫
      pullback.fst (tensorObj (𝒞 := instCategoryOver) R S).hom f ≫ pullback.snd R.hom S.hom)
    (pullback.snd ((pullback f).obj R).hom ((pullback f).obj S).hom ≫ pullback.fst S.hom f)
have rw : Hom.left (𝟙 ((pullback f).obj (tensorObj (𝒞 := instCategoryOver) R S))) ≫
    pullback.fst (tensorObj (𝒞 := instCategoryOver) R S).hom f ≫ pullback.snd R.hom S.hom =
  CategoryStruct.comp (Y := (tensorObj (𝒞 := instCategoryOver) ((pullback f).obj R) ((pullback f).obj S)).left)
    (Hom.left (Iso.hom (prodComparisonIso (pullback f) R S)))
    (pullback.snd ((pullback f).obj R).hom ((pullback f).obj S).hom ≫ pullback.fst S.hom f)
Tactic: rw [Monoidal.μ_of_cartesianMonoidalCategory,
    ← cancel_epi (prodComparisonIso (Over.pullback f) R S).hom.left,
    ← Over.comp_left_assoc, Iso.hom_inv_id]
-- Lean 4 proof step from Mathlib.CategoryTheory.Monoidal.Cartesian.Over (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : HasPullbacks.{v_1, u_1} C
X R_1 : C
f : R_1 ⟶ X
R S : Over.{v_1, u_1} X
⊢ Eq (α := (tensorObj (𝒞 := instCategoryOver) ((pullback f).obj R) ((pullback f).obj S)).left ⟶ R_1)
    (Hom.left (LaxMonoidal.μ (pullback f) R S) ≫ pullback.snd (tensorObj (𝒞 := instCategoryOver) R S).hom f)
    (pullback.snd ((pullback f).obj R).hom ((pullback f).obj S).hom ≫ pullback.snd S.hom f)
have rw : Hom.left (𝟙 ((pullback f).obj (tensorObj (𝒞 := instCategoryOver) R S))) ≫
    pullback.snd (tensorObj (𝒞 := instCategoryOver) R S).hom f =
  CategoryStruct.comp (Y := (tensorObj (𝒞 := instCategoryOver) ((pullback f).obj R) ((pullback f).obj S)).left)
    (Hom.left (Iso.hom (prodComparisonIso (pullback f) R S)))
    (pullback.snd ((pullback f).obj R).hom ((pullback f).obj S).hom ≫ pullback.snd S.hom f)
Tactic: rw [Monoidal.μ_of_cartesianMonoidalCategory,
    ← cancel_epi (prodComparisonIso (Over.pullback f) R S).hom.left,
    ← Over.comp_left_assoc, Iso.hom_inv_id]
-- Lean 4 proof step from Mathlib.CategoryTheory.Monoidal.Cartesian.Over (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : HasPullbacks.{v_1, u_1} C
R S : C
f : R ⟶ S
⊢ Eq (α :=
    (pullback f).obj ((𝟙_ (Over.{v_1, u_1} S)) (𝒞 := instCategoryOver)) ≅
      (𝟙_ (Over.{v_1, u_1} R)) (𝒞 := instCategoryOver))
    (preservesTerminalIso (pullback f)) (isoMk (asIso (pullback.snd (𝟙 S) f)) ⋯)
have w : Eq (α :=
  (pullback f).obj ((𝟙_ (Over.{v_1, u_1} S)) (𝒞 := instCategoryOver)) ⟶
    (𝟙_ (Over.{v_1, u_1} R)) (𝒞 := instCategoryOver))
  (Iso.hom (preservesTerminalIso (pullback f))) (Iso.hom (isoMk (asIso (pullback.snd (𝟙 S) f)) sorry))
Tactic: ext1
-- Lean 4 proof step from Mathlib.CategoryTheory.Monoidal.Cartesian.Over (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : HasPullbacks.{v_1, u_1} C
X Y : C
f : X ⟶ Y
A B : Over.{v_1, u_1} Y
⊢ Eq (α := (tensorObj (𝒞 := instCategoryOver) ((pullback f).obj A) ((pullback f).obj B)).left ⟶ A.left)
    (CategoryStruct.comp (Y := ((pullback f).obj (tensorObj (𝒞 := instCategoryOver) A B)).left)
      (Hom.left (Iso.inv (prodComparisonIso (pullback f) A B)))
      (pullback.fst (pullback.fst A.hom B.hom ≫ A.hom) f ≫ pullback.fst A.hom B.hom))
    (pullback.fst (pullback.snd A.hom f) (pullback.snd B.hom f) ≫ pullback.fst A.hom f)
have rw : Eq (α := ((pullback f).obj (tensorObj (𝒞 := instCategoryOver) A B)).left ⟶ A.left)
  (pullback.fst (pullback.fst A.hom B.hom ≫ A.hom) f ≫ pullback.fst A.hom B.hom)
  (CategoryStruct.comp (Y := (tensorObj (𝒞 := instCategoryOver) ((pullback f).obj A) ((pullback f).obj B)).left)
    (Hom.left (Iso.hom (prodComparisonIso (pullback f) A B)))
    (pullback.fst (pullback.snd A.hom f) (pullback.snd B.hom f) ≫ pullback.fst A.hom f))
Tactic: rw [← cancel_epi (prodComparisonIso (Over.pullback f) A B).hom.left,
    Over.hom_left_inv_left_assoc]
-- Lean 4 proof step from Mathlib.CategoryTheory.Monoidal.Cartesian.Over (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : HasPullbacks.{v_1, u_1} C
X A B Y : C
f : X ⟶ Y
gA : A ⟶ Y
gB : B ⟶ Y
⊢ Eq (α := (tensorObj (𝒞 := instCategoryOver) ((pullback f).obj (mk gA)) ((pullback f).obj (mk gB))).left ⟶ B)
    (CategoryStruct.comp (Y := ((pullback f).obj (tensorObj (𝒞 := instCategoryOver) (mk gA) (mk gB))).left)
      (Hom.left (Iso.inv (prodComparisonIso (pullback f) (mk gA) (mk gB))))
      (pullback.fst (pullback.fst gA gB ≫ gA) f ≫ pullback.snd gA gB))
    (pullback.snd ((pullback f).obj (mk gA)).hom ((pullback f).obj (mk gB)).hom ≫ pullback.fst (mk gB).hom f)
have rw : Eq (α := ((pullback f).obj (tensorObj (𝒞 := instCategoryOver) (mk gA) (mk gB))).left ⟶ B)
  (pullback.fst (pullback.fst gA gB ≫ gA) f ≫ pullback.snd gA gB)
  (CategoryStruct.comp (Y :=
    (tensorObj (𝒞 := instCategoryOver) ((pullback f).obj (mk gA)) ((pullback f).obj (mk gB))).left)
    (Hom.left (Iso.hom (prodComparisonIso (pullback f) (mk gA) (mk gB))))
    (pullback.snd ((pullback f).obj (mk gA)).hom ((pullback f).obj (mk gB)).hom ≫ pullback.fst (mk gB).hom f))
Tactic: rw [← cancel_epi (prodComparisonIso (Over.pullback f) _ _).hom.left,
    Over.hom_left_inv_left_assoc]
-- Lean 4 proof step from Mathlib.CategoryTheory.Monoidal.Cartesian.Over (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : HasPullbacks.{v_1, u_1} C
X A B Y : C
f : X ⟶ Y
gA : A ⟶ Y
gB : B ⟶ Y
⊢ Eq (α := (tensorObj (𝒞 := instCategoryOver) ((pullback f).obj (mk gA)) ((pullback f).obj (mk gB))).left ⟶ X)
    (CategoryStruct.comp (Y := ((pullback f).obj (tensorObj (𝒞 := instCategoryOver) (mk gA) (mk gB))).left)
      (Hom.left (Iso.inv (prodComparisonIso (pullback f) (mk gA) (mk gB)))) (pullback.snd (pullback.fst gA gB ≫ gA) f))
    (pullback.snd ((pullback f).obj (mk gA)).hom ((pullback f).obj (mk gB)).hom ≫ pullback.snd (mk gB).hom f)
have rw : Eq (α := ((pullback f).obj (tensorObj (𝒞 := instCategoryOver) (mk gA) (mk gB))).left ⟶ X)
  (pullback.snd (pullback.fst gA gB ≫ gA) f)
  (CategoryStruct.comp (Y :=
    (tensorObj (𝒞 := instCategoryOver) ((pullback f).obj (mk gA)) ((pullback f).obj (mk gB))).left)
    (Hom.left (Iso.hom (prodComparisonIso (pullback f) (mk gA) (mk gB))))
    (pullback.snd ((pullback f).obj (mk gA)).hom ((pullback f).obj (mk gB)).hom ≫ pullback.snd (mk gB).hom f))
Tactic: rw [← cancel_epi (prodComparisonIso (Over.pullback f) _ _).hom.left,
    Over.hom_left_inv_left_assoc]
-- Lean 4 proof step from Mathlib.CategoryTheory.Monoidal.Cartesian.Over (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : HasPullbacks.{v_1, u_1} C
X A B R S Y Z : C
f : R ⟶ X
g : S ⟶ X
inst_2 : MonObj (mk f)
⊢ Eq.{v_1 + 1} (α :=
    tensorObj (𝒞 := instCategoryOver) (mk (Y := Limits.pullback f (𝟙 X)) (pullback.snd f (𝟙 X)))
        (mk (Y := Limits.pullback f (𝟙 X)) (pullback.snd f (𝟙 X))) ⟶
      mk f)
    (MonObj.mul ≫ homMk (pullback.fst f (𝟙 X)) ⋯)
    ((homMk (pullback.fst f (𝟙 X)) ⋯ ⊗ₘ homMk (pullback.fst f (𝟙 X)) ⋯) ≫ MonObj.mul)
have h : Eq.{v_1 + 1} (α :=
  (tensorObj (𝒞 := instCategoryOver) (mk (Y := Limits.pullback f (𝟙 X)) (pullback.snd f (𝟙 X)))
        (mk (Y := Limits.pullback f (𝟙 X)) (pullback.snd f (𝟙 X)))).left ⟶
    (mk f).left)
  (Hom.left (MonObj.mul ≫ homMk (pullback.fst f (𝟙 X)) sorry))
  (Hom.left ((homMk (pullback.fst f (𝟙 X)) sorry ⊗ₘ homMk (pullback.fst f (𝟙 X)) sorry) ≫ MonObj.mul))
Tactic: ext