awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.CategoryTheory.Monoidal.Internal.Types.Basic (goal)
Goal:
A : CommMonCat
⊢ Iso.hom
        (β_
          (MonTypeEquivalenceMon.inverse.obj
              (Functor.obj
                (forget₂ CommMonCat (FC := fun (x1 x2 : CommMonCat.{u}) ↦ (↑x1 : Type u) →* (↑x2 : Type u)) (CC :=
                  CommMonCat.carrier) MonCat (FD := fun (x1 x2 : MonCat.{u}) ↦ (↑x1 : Type u) →* (↑x2 : Type u)) (CD :=
                  MonCat.carrier))
                A)).X
          (MonTypeEquivalenceMon.inverse.obj
              (Functor.obj
                (forget₂ CommMonCat (FC := fun (x1 x2 : CommMonCat.{u}) ↦ (↑x1 : Type u) →* (↑x2 : Type u)) (CC :=
                  CommMonCat.carrier) MonCat (FD := fun (x1 x2 : MonCat.{u}) ↦ (↑x1 : Type u) →* (↑x2 : Type u)) (CD :=
                  MonCat.carrier))
                A)).X) ≫
      μ =
    μ
have h : ∀ (x y : (↑A : Type u)),
  (hom (FC := TypeCat.Fun)
          (Iso.hom
              (β_
                (MonTypeEquivalenceMon.inverse.obj
                    (Functor.obj
                      (forget₂ CommMonCat (FC := fun (x1 x2 : CommMonCat.{u}) ↦ (↑x1 : Type u) →* (↑x2 : Type u)) (CC :=
                        CommMonCat.carrier) MonCat (FD := fun (x1 x2 : MonCat.{u}) ↦ (↑x1 : Type u) →* (↑x2 : Type u))
                        (CD := MonCat.carrier))
                      A)).X
                (MonTypeEquivalenceMon.inverse.obj
                    (Functor.obj
                      (forget₂ CommMonCat (FC := fun (x1 x2 : CommMonCat.{u}) ↦ (↑x1 : Type u) →* (↑x2 : Type u)) (CC :=
                        CommMonCat.carrier) MonCat (FD := fun (x1 x2 : MonCat.{u}) ↦ (↑x1 : Type u) →* (↑x2 : Type u))
                        (CD := MonCat.carrier))
                      A)).X) ≫
            μ)).toFun
      (x, y) =
    TypeCat.Fun.toFun (hom (FC := TypeCat.Fun) μ) (x, y)
Tactic: ext ⟨x : A, y : A⟩
-- Lean 4 proof step from Mathlib.CategoryTheory.Monoidal.Internal.Types.CommGrp_ (goal)
Goal:
A : CommGrpCat
⊢ Iso.hom
        (β_
          (grpTypeEquivalenceGrp.inverse.obj
              (Functor.obj
                (forget₂ CommGrpCat (FC := fun (x1 x2 : CommGrpCat.{u}) ↦ (↑x1 : Type u) →* (↑x2 : Type u)) (CC :=
                  CommGrpCat.carrier) GrpCat (FD := fun (x1 x2 : GrpCat.{u}) ↦ (↑x1 : Type u) →* (↑x2 : Type u)) (CD :=
                  GrpCat.carrier))
                A)).X
          (grpTypeEquivalenceGrp.inverse.obj
              (Functor.obj
                (forget₂ CommGrpCat (FC := fun (x1 x2 : CommGrpCat.{u}) ↦ (↑x1 : Type u) →* (↑x2 : Type u)) (CC :=
                  CommGrpCat.carrier) GrpCat (FD := fun (x1 x2 : GrpCat.{u}) ↦ (↑x1 : Type u) →* (↑x2 : Type u)) (CD :=
                  GrpCat.carrier))
                A)).X) ≫
      μ =
    μ
have h : ∀ (x y : (↑A : Type u)),
  (hom (FC := TypeCat.Fun)
          (Iso.hom
              (β_
                (grpTypeEquivalenceGrp.inverse.obj
                    (Functor.obj
                      (forget₂ CommGrpCat (FC := fun (x1 x2 : CommGrpCat.{u}) ↦ (↑x1 : Type u) →* (↑x2 : Type u)) (CC :=
                        CommGrpCat.carrier) GrpCat (FD := fun (x1 x2 : GrpCat.{u}) ↦ (↑x1 : Type u) →* (↑x2 : Type u))
                        (CD := GrpCat.carrier))
                      A)).X
                (grpTypeEquivalenceGrp.inverse.obj
                    (Functor.obj
                      (forget₂ CommGrpCat (FC := fun (x1 x2 : CommGrpCat.{u}) ↦ (↑x1 : Type u) →* (↑x2 : Type u)) (CC :=
                        CommGrpCat.carrier) GrpCat (FD := fun (x1 x2 : GrpCat.{u}) ↦ (↑x1 : Type u) →* (↑x2 : Type u))
                        (CD := GrpCat.carrier))
                      A)).X) ≫
            μ)).toFun
      (x, y) =
    TypeCat.Fun.toFun (hom (FC := TypeCat.Fun) μ) (x, y)
Tactic: ext ⟨x : A, y : A⟩
-- Lean 4 proof step from Mathlib.CategoryTheory.Monoidal.Internal.Types.Grp (goal)
Goal:
A : GrpCat
⊢ CartesianMonoidalCategory.lift (↾fun (x : (↑A : Type u)) ↦ x⁻¹)
        (𝟙
          (MonTypeEquivalenceMon.inverse.obj
              (Functor.obj
                (forget₂ GrpCat (FC := fun (x1 x2 : GrpCat.{u}) ↦ (↑x1 : Type u) →* (↑x2 : Type u)) (CC :=
                  GrpCat.carrier) MonCat (FD := fun (x1 x2 : MonCat.{u}) ↦ (↑x1 : Type u) →* (↑x2 : Type u)) (CD :=
                  MonCat.carrier))
                A)).X) ≫
      μ =
    SemiCartesianMonoidalCategory.toUnit
        (MonTypeEquivalenceMon.inverse.obj
            (Functor.obj
              (forget₂ GrpCat (FC := fun (x1 x2 : GrpCat.{u}) ↦ (↑x1 : Type u) →* (↑x2 : Type u)) (CC := GrpCat.carrier)
                MonCat (FD := fun (x1 x2 : MonCat.{u}) ↦ (↑x1 : Type u) →* (↑x2 : Type u)) (CD := MonCat.carrier))
              A)).X ≫
      η
have h : ∀
  (x :
    (MonTypeEquivalenceMon.inverse.obj
        (Functor.obj
          (forget₂ GrpCat (FC := fun (x1 x2 : GrpCat.{u}) ↦ (↑x1 : Type u) →* (↑x2 : Type u)) (CC := GrpCat.carrier)
            MonCat (FD := fun (x1 x2 : MonCat.{u}) ↦ (↑x1 : Type u) →* (↑x2 : Type u)) (CD := MonCat.carrier))
          A)).X),
  (ConcreteCategory.hom (FC := TypeCat.Fun)
          (CartesianMonoidalCategory.lift (↾fun (x : (↑A : Type u)) ↦ x⁻¹)
              (𝟙
                (MonTypeEquivalenceMon.inverse.obj
                    (Functor.obj
                      (forget₂ GrpCat (FC := fun (x1 x2 : GrpCat.{u}) ↦ (↑x1 : Type u) →* (↑x2 : Type u)) (CC :=
                        GrpCat.carrier) MonCat (FD := fun (x1 x2 : MonCat.{u}) ↦ (↑x1 : Type u) →* (↑x2 : Type u))
                        (CD := MonCat.carrier))
                      A)).X) ≫
            μ)).toFun
      x =
    TypeCat.Fun.toFun
      (ConcreteCategory.hom (FC := TypeCat.Fun)
        (SemiCartesianMonoidalCategory.toUnit
            (MonTypeEquivalenceMon.inverse.obj
                (Functor.obj
                  (forget₂ GrpCat (FC := fun (x1 x2 : GrpCat.{u}) ↦ (↑x1 : Type u) →* (↑x2 : Type u)) (CC :=
                    GrpCat.carrier) MonCat (FD := fun (x1 x2 : MonCat.{u}) ↦ (↑x1 : Type u) →* (↑x2 : Type u)) (CD :=
                    MonCat.carrier))
                  A)).X ≫
          η))
      x
Tactic: ext x
-- Lean 4 proof step from Mathlib.CategoryTheory.Monoidal.Internal.Types.Grp (goal)
Goal:
A : GrpCat
⊢ CartesianMonoidalCategory.lift
        (𝟙
          (MonTypeEquivalenceMon.inverse.obj
              (Functor.obj
                (forget₂ GrpCat (FC := fun (x1 x2 : GrpCat.{u}) ↦ (↑x1 : Type u) →* (↑x2 : Type u)) (CC :=
                  GrpCat.carrier) MonCat (FD := fun (x1 x2 : MonCat.{u}) ↦ (↑x1 : Type u) →* (↑x2 : Type u)) (CD :=
                  MonCat.carrier))
                A)).X)
        (↾fun (x : (↑A : Type u)) ↦ x⁻¹) ≫
      μ =
    SemiCartesianMonoidalCategory.toUnit
        (MonTypeEquivalenceMon.inverse.obj
            (Functor.obj
              (forget₂ GrpCat (FC := fun (x1 x2 : GrpCat.{u}) ↦ (↑x1 : Type u) →* (↑x2 : Type u)) (CC := GrpCat.carrier)
                MonCat (FD := fun (x1 x2 : MonCat.{u}) ↦ (↑x1 : Type u) →* (↑x2 : Type u)) (CD := MonCat.carrier))
              A)).X ≫
      η
have h : ∀
  (x :
    (MonTypeEquivalenceMon.inverse.obj
        (Functor.obj
          (forget₂ GrpCat (FC := fun (x1 x2 : GrpCat.{u}) ↦ (↑x1 : Type u) →* (↑x2 : Type u)) (CC := GrpCat.carrier)
            MonCat (FD := fun (x1 x2 : MonCat.{u}) ↦ (↑x1 : Type u) →* (↑x2 : Type u)) (CD := MonCat.carrier))
          A)).X),
  (ConcreteCategory.hom (FC := TypeCat.Fun)
          (CartesianMonoidalCategory.lift
              (𝟙
                (MonTypeEquivalenceMon.inverse.obj
                    (Functor.obj
                      (forget₂ GrpCat (FC := fun (x1 x2 : GrpCat.{u}) ↦ (↑x1 : Type u) →* (↑x2 : Type u)) (CC :=
                        GrpCat.carrier) MonCat (FD := fun (x1 x2 : MonCat.{u}) ↦ (↑x1 : Type u) →* (↑x2 : Type u))
                        (CD := MonCat.carrier))
                      A)).X)
              (↾fun (x : (↑A : Type u)) ↦ x⁻¹) ≫
            μ)).toFun
      x =
    TypeCat.Fun.toFun
      (ConcreteCategory.hom (FC := TypeCat.Fun)
        (SemiCartesianMonoidalCategory.toUnit
            (MonTypeEquivalenceMon.inverse.obj
                (Functor.obj
                  (forget₂ GrpCat (FC := fun (x1 x2 : GrpCat.{u}) ↦ (↑x1 : Type u) →* (↑x2 : Type u)) (CC :=
                    GrpCat.carrier) MonCat (FD := fun (x1 x2 : MonCat.{u}) ↦ (↑x1 : Type u) →* (↑x2 : Type u)) (CD :=
                    MonCat.carrier))
                  A)).X ≫
          η))
      x
Tactic: ext x
-- Lean 4 proof step from Mathlib.CategoryTheory.Monoidal.Limits.Basic (goal)
Goal:
J : Type w
inst : SmallCategory J
C : Type u
inst_1 : Category.{v, u} C
inst_2 : HasLimitsOfShape.{w, w, v, u} J C
inst_3 : MonoidalCategory.{v, u} C
F G : Functor.{w, v, w, u} J C
j j' : J
f : j ⟶ j'
⊢ Eq.{v + 1} (α :=
    Functor.obj.{w, v, w, u} (Functor.obj (Functor.const J) (tensorObj.{v, u} (𝒞 := inst_1) (limit F) (limit G))) j ⟶
      (tensorObj (𝒞 := Functor.category) F G).obj j')
    (Functor.map (Functor.obj (Functor.const J) (tensorObj.{v, u} (𝒞 := inst_1) (limit F) (limit G))) f ≫
      (limit.π F j' ⊗ₘ limit.π G j'))
    ((limit.π F j ⊗ₘ limit.π G j) ≫ (tensorObj (𝒞 := Functor.category) F G).map f)
have dsimp : 𝟙 (tensorObj.{v, u} (𝒞 := inst_1) (limit F) (limit G)) ≫ (limit.π F j' ⊗ₘ limit.π G j') =
  (limit.π F j ⊗ₘ limit.π G j) ≫ (F.map f ⊗ₘ G.map f)
Tactic: dsimp
-- Lean 4 proof step from Mathlib.CategoryTheory.Monoidal.Limits.Basic (goal)
Goal:
J : Type w
inst : SmallCategory J
C : Type u
inst_1 : Category.{v, u} C
inst_2 : HasLimitsOfShape.{w, w, v, u} J C
inst_3 : MonoidalCategory.{v, u} C
F G H : Functor.{w, v, w, u} J C
j : J
| (CategoryStruct.comp (Y := tensorObj.{v, u} (𝒞 := inst_1) (limit (tensorObj (𝒞 := Functor.category) F G)) (limit H))
      (limit.lift (tensorObj (𝒞 := Functor.category) F G)
          { pt := tensorObj.{v, u} (𝒞 := inst_1) (limit F) (limit G),
            π := { app := fun (j : J) ↦ limit.π F j ⊗ₘ limit.π G j, naturality := ⋯ } } ▷
        limit H)
      ((limit.π (tensorObj (𝒞 := Functor.category) F G) j ⊗ₘ limit.π H j) ≫
        Iso.hom (α_ (F.obj j) (G.obj j) (H.obj j))) :
    tensorObj.{v, u} (𝒞 := inst_1) (tensorObj.{v, u} (𝒞 := inst_1) (limit F) (limit G)) (limit H) ⟶
      tensorObj.{v, u} (𝒞 := inst_1) (F.obj j) (tensorObj.{v, u} (𝒞 := inst_1) (G.obj j) (H.obj j)))
have rw : ((NatTrans.app
        (Cone.π
          { pt := tensorObj.{v, u} (𝒞 := inst_1) (limit F) (limit G),
            π := { app := fun (j : J) ↦ limit.π F j ⊗ₘ limit.π G j, naturality := sorry } })
        j ⊗ₘ
      limit.π H j) ≫
    Iso.hom (α_ (F.obj j) (G.obj j) (H.obj j)) :
  tensorObj.{v, u} (𝒞 := inst_1) (tensorObj.{v, u} (𝒞 := inst_1) (limit F) (limit G)) (limit H) ⟶
    tensorObj.{v, u} (𝒞 := inst_1) (F.obj j) (tensorObj.{v, u} (𝒞 := inst_1) (G.obj j) (H.obj j)))
Tactic: rw [tensorHom_def, Category.assoc, ← comp_whiskerRight_assoc,
        limit.lift_π, tensor_whiskerLeft, Category.assoc, Category.assoc,
        Iso.inv_hom_id, Category.comp_id,
        ← associator_naturality_right, ← tensorHom_def_assoc]
-- Lean 4 proof step from Mathlib.CategoryTheory.Monoidal.Limits.Basic (goal)
Goal:
J : Type w
inst : SmallCategory J
C : Type u
inst_1 : Category.{v, u} C
inst_2 : HasLimitsOfShape.{w, w, v, u} J C
inst_3 : MonoidalCategory.{v, u} C
F G H : Functor.{w, v, w, u} J C
j : J
⊢ Eq (α :=
    tensorObj.{v, u} (𝒞 := inst_1) (tensorObj.{v, u} (𝒞 := inst_1) (limit F) (limit G)) (limit H) ⟶
      tensorObj.{v, u} (𝒞 := inst_1) (F.obj j) (tensorObj.{v, u} (𝒞 := inst_1) (G.obj j) (H.obj j)))
    ((NatTrans.app
            (Cone.π
              { pt := tensorObj.{v, u} (𝒞 := inst_1) (limit F) (limit G),
                π := { app := fun (j : J) ↦ limit.π F j ⊗ₘ limit.π G j, naturality := ⋯ } })
            j ⊗ₘ
          limit.π H j) ≫
        Iso.hom (α_ (F.obj j) (G.obj j) (H.obj j)) :
      tensorObj.{v, u} (𝒞 := inst_1) (tensorObj.{v, u} (𝒞 := inst_1) (limit F) (limit G)) (limit H) ⟶
        tensorObj.{v, u} (𝒞 := inst_1) (F.obj j) (tensorObj.{v, u} (𝒞 := inst_1) (G.obj j) (H.obj j)))
    (Iso.hom (α_ (limit F) (limit G) (limit H)) ≫
        CategoryStruct.comp (Y :=
          tensorObj.{v, u} (𝒞 := inst_1) (limit F) (limit (tensorObj (𝒞 := Functor.category) G H)))
          (limit F ◁
            limit.lift (tensorObj (𝒞 := Functor.category) G H)
              { pt := tensorObj.{v, u} (𝒞 := inst_1) (limit G) (limit H),
                π := { app := fun (j : J) ↦ limit.π G j ⊗ₘ limit.π H j, naturality := ⋯ } })
          (limit.π F j ⊗ₘ limit.π (tensorObj (𝒞 := Functor.category) G H) j) :
      tensorObj.{v, u} (𝒞 := inst_1) (tensorObj.{v, u} (𝒞 := inst_1) (limit F) (limit G)) (limit H) ⟶
        tensorObj.{v, u} (𝒞 := inst_1) (F.obj j) (tensorObj.{v, u} (𝒞 := inst_1) (G.obj j) (H.obj j)))
have dsimp : Eq (α :=
  tensorObj.{v, u} (𝒞 := inst_1) (tensorObj.{v, u} (𝒞 := inst_1) (limit F) (limit G)) (limit H) ⟶
    tensorObj.{v, u} (𝒞 := inst_1) (F.obj j) (tensorObj.{v, u} (𝒞 := inst_1) (G.obj j) (H.obj j)))
  (((limit.π F j ⊗ₘ limit.π G j) ⊗ₘ limit.π H j) ≫ Iso.hom (α_ (F.obj j) (G.obj j) (H.obj j)))
  (Iso.hom (α_ (limit F) (limit G) (limit H)) ≫
    CategoryStruct.comp (Y := tensorObj.{v, u} (𝒞 := inst_1) (limit F) (limit (tensorObj (𝒞 := Functor.category) G H)))
      (limit F ◁
        limit.lift (tensorObj (𝒞 := Functor.category) G H)
          { pt := tensorObj.{v, u} (𝒞 := inst_1) (limit G) (limit H),
            π := { app := fun (j : J) ↦ limit.π G j ⊗ₘ limit.π H j, naturality := sorry } })
      (limit.π F j ⊗ₘ limit.π (tensorObj (𝒞 := Functor.category) G H) j))
Tactic: dsimp
-- Lean 4 proof step from Mathlib.CategoryTheory.Monoidal.Limits.Basic (goal)
Goal:
J : Type w
inst : SmallCategory J
C : Type u
inst_1 : Category.{v, u} C
inst_2 : HasLimitsOfShape.{w, w, v, u} J C
inst_3 : MonoidalCategory.{v, u} C
F G H : Functor.{w, v, w, u} J C
j : J
| Iso.hom (α_ (limit F) (limit G) (limit H)) ≫
    CategoryStruct.comp (Y := tensorObj.{v, u} (𝒞 := inst_1) (limit F) (limit (tensorObj (𝒞 := Functor.category) G H)))
      (limit F ◁
        limit.lift (tensorObj (𝒞 := Functor.category) G H)
          { pt := tensorObj.{v, u} (𝒞 := inst_1) (limit G) (limit H),
            π := { app := fun (j : J) ↦ limit.π G j ⊗ₘ limit.π H j, naturality := ⋯ } })
      (limit.π F j ⊗ₘ limit.π (tensorObj (𝒞 := Functor.category) G H) j)
have rw : limit.π F j ▷ limit G ▷ limit H ≫
  Iso.hom (α_ (F.obj j) (limit G) (limit H)) ≫
    F.obj j ◁
      NatTrans.app
        (Cone.π
          { pt := tensorObj.{v, u} (𝒞 := inst_1) (limit G) (limit H),
            π := { app := fun (j : J) ↦ limit.π G j ⊗ₘ limit.π H j, naturality := sorry } })
        j
Tactic: rw [tensorHom_def, ← whisker_exchange,
        ← whiskerLeft_comp_assoc, limit.lift_π,
        whisker_exchange, ← associator_naturality_left_assoc]
-- Lean 4 proof step from Mathlib.CategoryTheory.Monoidal.Limits.Basic (goal)
Goal:
J : Type w
inst : SmallCategory J
C : Type u
inst_1 : Category.{v, u} C
inst_2 : HasLimitsOfShape.{w, w, v, u} J C
inst_3 : MonoidalCategory.{v, u} C
F G H : Functor.{w, v, w, u} J C
j : J
⊢ Eq (α :=
    tensorObj.{v, u} (𝒞 := inst_1) (tensorObj.{v, u} (𝒞 := inst_1) (limit F) (limit G)) (limit H) ⟶
      tensorObj.{v, u} (𝒞 := inst_1) (F.obj j) (tensorObj.{v, u} (𝒞 := inst_1) (G.obj j) (H.obj j)))
    (((limit.π F j ⊗ₘ limit.π G j) ⊗ₘ limit.π H j) ≫ Iso.hom (α_ (F.obj j) (G.obj j) (H.obj j)))
    (Iso.hom (α_ (limit F) (limit G) (limit H)) ≫
      CategoryStruct.comp (Y :=
        tensorObj.{v, u} (𝒞 := inst_1) (limit F) (limit (tensorObj (𝒞 := Functor.category) G H)))
        (limit F ◁
          limit.lift (tensorObj (𝒞 := Functor.category) G H)
            { pt := tensorObj.{v, u} (𝒞 := inst_1) (limit G) (limit H),
              π := { app := fun (j : J) ↦ limit.π G j ⊗ₘ limit.π H j, naturality := ⋯ } })
        (limit.π F j ⊗ₘ limit.π (tensorObj (𝒞 := Functor.category) G H) j))
have conv_rhs : Eq (α :=
  tensorObj.{v, u} (𝒞 := inst_1) (tensorObj.{v, u} (𝒞 := inst_1) (limit F) (limit G)) (limit H) ⟶
    tensorObj.{v, u} (𝒞 := inst_1) (F.obj j) (tensorObj.{v, u} (𝒞 := inst_1) (G.obj j) (H.obj j)))
  (((limit.π F j ⊗ₘ limit.π G j) ⊗ₘ limit.π H j) ≫ Iso.hom (α_ (F.obj j) (G.obj j) (H.obj j)))
  (limit.π F j ▷ limit G ▷ limit H ≫
    Iso.hom (α_ (F.obj j) (limit G) (limit H)) ≫
      F.obj j ◁
        NatTrans.app
          (Cone.π
            { pt := tensorObj.{v, u} (𝒞 := inst_1) (limit G) (limit H),
              π := { app := fun (j : J) ↦ limit.π G j ⊗ₘ limit.π H j, naturality := sorry } })
          j)
Tactic: conv_rhs => rw [tensorHom_def, ← whisker_exchange,
        ← whiskerLeft_comp_assoc, limit.lift_π,
        whisker_exchange, ← associator_naturality_left_assoc]
-- Lean 4 proof step from Mathlib.CategoryTheory.Monoidal.Limits.Basic (goal)
Goal:
J : Type w
inst : SmallCategory J
C : Type u
inst_1 : Category.{v, u} C
inst_2 : HasLimitsOfShape.{w, w, v, u} J C
inst_3 : MonoidalCategory.{v, u} C
F G H : Functor.{w, v, w, u} J C
j : J
⊢ Eq (α :=
    tensorObj.{v, u} (𝒞 := inst_1) (tensorObj.{v, u} (𝒞 := inst_1) (limit F) (limit G)) (limit H) ⟶
      tensorObj.{v, u} (𝒞 := inst_1) (F.obj j) (tensorObj.{v, u} (𝒞 := inst_1) (G.obj j) (H.obj j)))
    (((limit.π F j ⊗ₘ limit.π G j) ⊗ₘ limit.π H j) ≫ Iso.hom (α_ (F.obj j) (G.obj j) (H.obj j)))
    (limit.π F j ▷ limit G ▷ limit H ≫
      Iso.hom (α_ (F.obj j) (limit G) (limit H)) ≫
        F.obj j ◁
          NatTrans.app
            (Cone.π
              { pt := tensorObj.{v, u} (𝒞 := inst_1) (limit G) (limit H),
                π := { app := fun (j : J) ↦ limit.π G j ⊗ₘ limit.π H j, naturality := ⋯ } })
            j)
have dsimp : Eq (α :=
  tensorObj.{v, u} (𝒞 := inst_1) (tensorObj.{v, u} (𝒞 := inst_1) (limit F) (limit G)) (limit H) ⟶
    tensorObj.{v, u} (𝒞 := inst_1) (F.obj j) (tensorObj.{v, u} (𝒞 := inst_1) (G.obj j) (H.obj j)))
  (((limit.π F j ⊗ₘ limit.π G j) ⊗ₘ limit.π H j) ≫ Iso.hom (α_ (F.obj j) (G.obj j) (H.obj j)))
  (limit.π F j ▷ limit G ▷ limit H ≫
    Iso.hom (α_ (F.obj j) (limit G) (limit H)) ≫ F.obj j ◁ (limit.π G j ⊗ₘ limit.π H j))
Tactic: dsimp only
-- Lean 4 proof step from Mathlib.CategoryTheory.Monoidal.Limits.Basic (goal)
Goal:
J : Type w
inst : SmallCategory J
C : Type u
inst_1 : Category.{v, u} C
inst_2 : HasLimitsOfShape.{w, w, v, u} J C
inst_3 : MonoidalCategory.{v, u} C
F G H : Functor.{w, v, w, u} J C
j : J
| limit.π F j ▷ limit G ▷ limit H ≫ Iso.hom (α_ (F.obj j) (limit G) (limit H)) ≫ F.obj j ◁ (limit.π G j ⊗ₘ limit.π H j)
have rw : ((limit.π F j ⊗ₘ limit.π G j) ⊗ₘ limit.π H j) ≫ Iso.hom (α_ (F.obj j) (G.obj j) (H.obj j))
Tactic: rw [tensorHom_def, whiskerLeft_comp,
        ← associator_naturality_middle_assoc,
        ← associator_naturality_right, ← comp_whiskerRight_assoc,
        ← tensorHom_def, ← tensorHom_def_assoc]
-- Lean 4 proof step from Mathlib.CategoryTheory.Monoidal.Limits.Basic (goal)
Goal:
J : Type w
inst : SmallCategory J
C : Type u
inst_1 : Category.{v, u} C
inst_2 : HasLimitsOfShape.{w, w, v, u} J C
inst_3 : MonoidalCategory.{v, u} C
F G H : Functor.{w, v, w, u} J C
j : J
⊢ Eq (α :=
    tensorObj.{v, u} (𝒞 := inst_1) (tensorObj.{v, u} (𝒞 := inst_1) (limit F) (limit G)) (limit H) ⟶
      tensorObj.{v, u} (𝒞 := inst_1) (F.obj j) (tensorObj.{v, u} (𝒞 := inst_1) (G.obj j) (H.obj j)))
    (((limit.π F j ⊗ₘ limit.π G j) ⊗ₘ limit.π H j) ≫ Iso.hom (α_ (F.obj j) (G.obj j) (H.obj j)))
    (limit.π F j ▷ limit G ▷ limit H ≫
      Iso.hom (α_ (F.obj j) (limit G) (limit H)) ≫ F.obj j ◁ (limit.π G j ⊗ₘ limit.π H j))
have conv_rhs : Eq (α :=
  tensorObj.{v, u} (𝒞 := inst_1) (tensorObj.{v, u} (𝒞 := inst_1) (limit F) (limit G)) (limit H) ⟶
    tensorObj.{v, u} (𝒞 := inst_1) (F.obj j) (tensorObj.{v, u} (𝒞 := inst_1) (G.obj j) (H.obj j)))
  (((limit.π F j ⊗ₘ limit.π G j) ⊗ₘ limit.π H j) ≫ Iso.hom (α_ (F.obj j) (G.obj j) (H.obj j)))
  (limit.π F j ▷ limit G ▷ limit H ≫
    Iso.hom (α_ (F.obj j) (limit G) (limit H)) ≫ F.obj j ◁ (limit.π G j ⊗ₘ limit.π H j))
Tactic: conv_rhs => rw [tensorHom_def, whiskerLeft_comp,
        ← associator_naturality_middle_assoc,
        ← associator_naturality_right, ← comp_whiskerRight_assoc,
        ← tensorHom_def, ← tensorHom_def_assoc]
-- Lean 4 proof step from Mathlib.CategoryTheory.Monoidal.Limits.Basic (goal)
Goal:
J : Type w
inst : SmallCategory J
C : Type u
inst_1 : Category.{v, u} C
inst_2 : HasLimitsOfShape.{w, w, v, u} J C
inst_3 : MonoidalCategory.{v, u} C
F G H : Functor.{w, v, w, u} J C
j : J
conv_rhs :
  Eq (α :=
    tensorObj.{v, u} (𝒞 := inst_1) (tensorObj.{v, u} (𝒞 := inst_1) (limit F) (limit G)) (limit H) ⟶
      tensorObj.{v, u} (𝒞 := inst_1) (F.obj j) (tensorObj.{v, u} (𝒞 := inst_1) (G.obj j) (H.obj j)))
    (((limit.π F j ⊗ₘ limit.π G j) ⊗ₘ limit.π H j) ≫ Iso.hom (α_ (F.obj j) (G.obj j) (H.obj j)))
    (limit.π F j ▷ limit G ▷ limit H ≫
      Iso.hom (α_ (F.obj j) (limit G) (limit H)) ≫ F.obj j ◁ (limit.π G j ⊗ₘ limit.π H j))
⊢ Eq (α :=
    tensorObj.{v, u} (𝒞 := inst_1) (tensorObj.{v, u} (𝒞 := inst_1) (limit F) (limit G)) (limit H) ⟶
      tensorObj.{v, u} (𝒞 := inst_1) (F.obj j) (tensorObj.{v, u} (𝒞 := inst_1) (G.obj j) (H.obj j)))
    (((limit.π F j ⊗ₘ limit.π G j) ⊗ₘ limit.π H j) ≫ Iso.hom (α_ (F.obj j) (G.obj j) (H.obj j)))
    (limit.π F j ▷ limit G ▷ limit H ≫
      Iso.hom (α_ (F.obj j) (limit G) (limit H)) ≫ F.obj j ◁ (limit.π G j ⊗ₘ limit.π H j))
have conv_rhs₁ : Eq (α :=
  tensorObj.{v, u} (𝒞 := inst_1) (tensorObj.{v, u} (𝒞 := inst_1) (limit F) (limit G)) (limit H) ⟶
    tensorObj.{v, u} (𝒞 := inst_1) (F.obj j) (tensorObj.{v, u} (𝒞 := inst_1) (G.obj j) (H.obj j)))
  (((limit.π F j ⊗ₘ limit.π G j) ⊗ₘ limit.π H j) ≫ Iso.hom (α_ (F.obj j) (G.obj j) (H.obj j)))
  (((limit.π F j ⊗ₘ limit.π G j) ⊗ₘ limit.π H j) ≫ Iso.hom (α_ (F.obj j) (G.obj j) (H.obj j)))
Tactic: conv_rhs => rw [tensorHom_def, whiskerLeft_comp,
        ← associator_naturality_middle_assoc,
        ← associator_naturality_right, ← comp_whiskerRight_assoc,
        ← tensorHom_def, ← tensorHom_def_assoc]