awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Module: Mathlib.CategoryTheory.Monoidal.Internal.Types.CommGrp_
Goal:
A : CommGrpCat
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)
⊢ 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

Tactic:
ext ⟨x : A, y : A⟩
-- Module: Mathlib.CategoryTheory.Monoidal.Internal.Types.CommGrp_
Goal:
A : CommGrpCat
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:
CommMonoid.mul_comm y x

Premises used: CommMonoid.mul_comm
-- Module: Mathlib.CategoryTheory.Monoidal.Internal.Types.CommGrp_
Goal:
A : CommGrpCat
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:
exact CommMonoid.mul_comm y x

Premises used: CommMonoid.mul_comm
-- Module: Mathlib.CategoryTheory.Monoidal.Internal.Types.CommGrp_
Goal:
A : CommGrpCat
⊢ (inverse.obj A).X = (↑A : Type u)

Tactic:
rfl

Premises used: rfl
-- Module: Mathlib.CategoryTheory.Monoidal.Internal.Types.CommGrp_
Goal:
A : CommGrpCat
x : (fun (X : Type u) ↦ X) (MonoidalCategoryStruct.tensorUnit (Type u) (𝒞 := types))
⊢ Eq (α := (↑A : Type u))
    ((hom (FC := TypeCat.Fun) η : (a : MonoidalCategoryStruct.tensorUnit (Type u)) → (↑A : Type u)) x : (↑A : Type u))
    (1 : (↑A : Type u))

Tactic:
rfl

Premises used: rfl
-- Module: Mathlib.CategoryTheory.Monoidal.Internal.Types.CommGrp_
Goal:
A : CommGrpCat
p : (fun (X : Type u) ↦ X) (MonoidalCategoryStruct.tensorObj (𝒞 := types) (inverse.obj A).X (inverse.obj A).X)
⊢ (hom (FC := TypeCat.Fun) μ : (a : MonoidalCategoryStruct.tensorObj (↑A : Type u) (↑A : Type u)) → (↑A : Type u)) p =
    p.1 * p.2

Tactic:
rfl

Premises used: rfl
-- Module: Mathlib.CategoryTheory.Monoidal.Internal.Types.CommGrp_
Goal:
A : CommGrpCat
x : (fun (X : Type u) ↦ X) (inverse.obj A).X
⊢ (hom (FC := TypeCat.Fun) (@GrpObj.inv _ _ typesCartesianMonoidalCategory _ _) : (a : (↑A : Type u)) → (↑A : Type u))
      x =
    x⁻¹

Tactic:
rfl

Premises used: rfl
-- Module: Mathlib.CategoryTheory.Monoidal.Internal.Types.Grp
Goal:
A : GrpCat
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
⊢ 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

Tactic:
ext x
-- Module: Mathlib.CategoryTheory.Monoidal.Internal.Types.Grp
Goal:
A : GrpCat
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:
inv_mul_cancel (G := A) x

Premises used: inv_mul_cancel
-- Module: Mathlib.CategoryTheory.Monoidal.Internal.Types.Grp
Goal:
A : GrpCat
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:
exact inv_mul_cancel (G := A) x

Premises used: inv_mul_cancel
-- Module: Mathlib.CategoryTheory.Monoidal.Internal.Types.Grp
Goal:
A : GrpCat
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
⊢ 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

Tactic:
ext x
-- Module: Mathlib.CategoryTheory.Monoidal.Internal.Types.Grp
Goal:
A : GrpCat
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:
mul_inv_cancel (G := A) x

Premises used: mul_inv_cancel
-- Module: Mathlib.CategoryTheory.Monoidal.Internal.Types.Grp
Goal:
A : GrpCat
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:
exact mul_inv_cancel (G := A) x

Premises used: mul_inv_cancel
-- Module: Mathlib.CategoryTheory.Monoidal.Limits.Basic
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'
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)
⊢ 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)

Tactic:
dsimp
-- Module: Mathlib.CategoryTheory.Monoidal.Limits.Basic
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'
⊢ 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:
simp only [Category.id_comp, tensorHom_comp_tensorHom, limit.w]

Premises used: CategoryTheory.MonoidalCategory.tensorHom_comp_tensorHom, CategoryTheory.Limits.limit.w, CategoryTheory.Category.id_comp
-- Module: Mathlib.CategoryTheory.Monoidal.Limits.Basic
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
X Y X' Y' : Functor.{w, v, w, u} J C
f : X ⟶ Y
g : X' ⟶ Y'
j : J
⊢ Eq (α := tensorObj.{v, u} (𝒞 := inst_1) (limit X) (limit X') ⟶ tensorObj.{v, u} (𝒞 := inst_1) (Y.obj j) (Y'.obj j))
    (CategoryStruct.comp (Y := limit (tensorObj (𝒞 := Functor.category) Y Y'))
      (CategoryStruct.comp (Y := tensorObj.{v, u} (𝒞 := inst_1) (limit Y) (limit Y')) (limMap f ⊗ₘ limMap g)
        (limit.lift (tensorObj (𝒞 := Functor.category) Y Y')
          { pt := tensorObj.{v, u} (𝒞 := inst_1) (limit Y) (limit Y'),
            π := { app := fun (j : J) ↦ limit.π Y j ⊗ₘ limit.π Y' j, naturality := ⋯ } }))
      (limit.π (tensorObj (𝒞 := Functor.category) Y Y') j))
    (CategoryStruct.comp (Y := limit (tensorObj (𝒞 := Functor.category) Y Y'))
      (CategoryStruct.comp (Y := limit (tensorObj (𝒞 := Functor.category) X X'))
        (limit.lift (tensorObj (𝒞 := Functor.category) X X')
          { pt := tensorObj.{v, u} (𝒞 := inst_1) (limit X) (limit X'),
            π := { app := fun (j : J) ↦ limit.π X j ⊗ₘ limit.π X' j, naturality := ⋯ } })
        (limMap (f ⊗ₘ g)))
      (limit.π (tensorObj (𝒞 := Functor.category) Y Y') j))

Tactic:
simp only [limit.lift_π, Cone.postcompose_obj_π, Monoidal.tensorHom_app, limit.lift_map,
        NatTrans.comp_app, Category.assoc, tensorHom_comp_tensorHom, limMap_π]

Premises used: CategoryTheory.Limits.limMap_π, CategoryTheory.Monoidal.tensorHom_app, CategoryTheory.Category.assoc, CategoryTheory.Limits.Cone.postcompose_obj_π, CategoryTheory.MonoidalCategory.tensorHom_comp_tensorHom, CategoryTheory.Limits.limit.lift_map, CategoryTheory.NatTrans.comp_app, CategoryTheory.Limits.limit.lift_π
-- Module: Mathlib.CategoryTheory.Monoidal.Limits.Basic
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
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 := ⋯ } })
          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)))
| (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)))

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]

Premises used: CategoryTheory.Category.assoc, CategoryTheory.Iso.inv_hom_id, CategoryTheory.MonoidalCategory.tensorHom_def, CategoryTheory.Category.comp_id, CategoryTheory.MonoidalCategory.comp_whiskerRight_assoc, CategoryTheory.MonoidalCategory.tensor_whiskerLeft, CategoryTheory.MonoidalCategory.tensorHom_def_assoc, CategoryTheory.Limits.limit.lift_π, CategoryTheory.MonoidalCategory.associator_naturality_right
-- Module: Mathlib.CategoryTheory.Monoidal.Limits.Basic
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
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 := ⋯ } })
        (limit.π F j ⊗ₘ limit.π (tensorObj (𝒞 := Functor.category) G 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)))
    ((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)))

Tactic:
dsimp