-- 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)))
    (CategoryStruct.comp (Y := limit (tensorObj (𝒞 := Functor.category) F (tensorObj (𝒞 := Functor.category) G H)))
        (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 := ⋯ } } ⊗ₘ
              𝟙 (limit H) :
            tensorObj.{v, u} (𝒞 := inst_1) (tensorObj.{v, u} (𝒞 := inst_1) (limit F) (limit G)) (limit H) ⟶
              tensorObj.{v, u} (𝒞 := inst_1) (limit (tensorObj (𝒞 := Functor.category) F G)) (limit H))
          (CategoryStruct.comp (Y :=
              limit (tensorObj (𝒞 := Functor.category) (tensorObj (𝒞 := Functor.category) F G) H))
              (limit.lift (tensorObj (𝒞 := Functor.category) (tensorObj (𝒞 := Functor.category) F G) H)
                { pt := tensorObj.{v, u} (𝒞 := inst_1) (limit (tensorObj (𝒞 := Functor.category) F G)) (limit H),
                  π :=
                    { app := fun (j : J) ↦ limit.π (tensorObj (𝒞 := Functor.category) F G) j ⊗ₘ limit.π H j,
                      naturality := ⋯ } })
              (limMap (Iso.hom (α_ F G H))) :
            tensorObj.{v, u} (𝒞 := inst_1) (limit (tensorObj (𝒞 := Functor.category) F G)) (limit H) ⟶
              limit (tensorObj (𝒞 := Functor.category) F (tensorObj (𝒞 := Functor.category) G H))))
        (limit.π (tensorObj (𝒞 := Functor.category) F (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)))
    (CategoryStruct.comp (Y := limit (tensorObj (𝒞 := Functor.category) F (tensorObj (𝒞 := Functor.category) G H)))
        (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)))
            (𝟙 (limit F) ⊗ₘ
                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 := ⋯ } } :
              tensorObj.{v, u} (𝒞 := inst_1) (limit F) (tensorObj.{v, u} (𝒞 := inst_1) (limit G) (limit H)) ⟶
                tensorObj.{v, u} (𝒞 := inst_1) (limit F) (limit (tensorObj (𝒞 := Functor.category) G H)))
            (limit.lift (tensorObj (𝒞 := Functor.category) F (tensorObj (𝒞 := Functor.category) G H))
                { pt := tensorObj.{v, u} (𝒞 := inst_1) (limit F) (limit (tensorObj (𝒞 := Functor.category) G H)),
                  π :=
                    { app := fun (j : J) ↦ limit.π F j ⊗ₘ limit.π (tensorObj (𝒞 := Functor.category) G H) j,
                      naturality := ⋯ } } :
              Cone.pt.{w, v, w, u} (J := J) (F :=
                  tensorObj (𝒞 := Functor.category) F (tensorObj (𝒞 := Functor.category) G H))
                  { pt := tensorObj.{v, u} (𝒞 := inst_1) (limit F) (limit (tensorObj (𝒞 := Functor.category) G H)),
                    π :=
                      { app := fun (j : J) ↦ limit.π F j ⊗ₘ limit.π (tensorObj (𝒞 := Functor.category) G H) j,
                        naturality := ⋯ :
                        NatTrans
                          (Functor.obj (Functor.const J)
                            (tensorObj.{v, u} (𝒞 := inst_1) (limit F) (limit (tensorObj (𝒞 := Functor.category) G H))))
                          (tensorObj (𝒞 := Functor.category) F (tensorObj (𝒞 := Functor.category) G H)) } :
                    Cone (tensorObj (𝒞 := Functor.category) F (tensorObj (𝒞 := Functor.category) G H)) } ⟶
                limit (tensorObj (𝒞 := Functor.category) F (tensorObj (𝒞 := Functor.category) G H))))
        (limit.π (tensorObj (𝒞 := Functor.category) F (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)))
⊢ Eq (α :=
    tensorObj.{v, u} (𝒞 := inst_1) (tensorObj.{v, u} (𝒞 := inst_1) (Functor.obj lim F) (Functor.obj lim G))
        (Functor.obj lim H) ⟶
      (tensorObj (𝒞 := Functor.category) F (tensorObj (𝒞 := Functor.category) G H)).obj j)
    (CategoryStruct.comp (Y := limit (tensorObj (𝒞 := Functor.category) F (tensorObj (𝒞 := Functor.category) G H)))
      (CategoryStruct.comp (Y :=
          tensorObj.{v, u} (𝒞 := inst_1) (Functor.obj lim (tensorObj (𝒞 := Functor.category) F G)) (Functor.obj lim H))
          ((fun (F G : Functor.{w, v, w, u} J C) ↦
                (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 := ⋯ } } :
                  Cone.pt.{w, v, w, u} (J := J) (F := 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 := ⋯ :
                            NatTrans
                              (Functor.obj (Functor.const J) (tensorObj.{v, u} (𝒞 := inst_1) (limit F) (limit G)))
                              (tensorObj (𝒞 := Functor.category) F G) } :
                        Cone (tensorObj (𝒞 := Functor.category) F G) } ⟶
                    limit (tensorObj (𝒞 := Functor.category) F G)))
              F G ⊗ₘ
            𝟙 (Functor.obj lim H))
          ((fun (F G : Functor.{w, v, w, u} J C) ↦
                (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 := ⋯ } } :
                  Cone.pt.{w, v, w, u} (J := J) (F := 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 := ⋯ :
                            NatTrans
                              (Functor.obj (Functor.const J) (tensorObj.{v, u} (𝒞 := inst_1) (limit F) (limit G)))
                              (tensorObj (𝒞 := Functor.category) F G) } :
                        Cone (tensorObj (𝒞 := Functor.category) F G) } ⟶
                    limit (tensorObj (𝒞 := Functor.category) F G)))
              (tensorObj (𝒞 := Functor.category) F G) H ≫
            Functor.map lim (Iso.hom (α_ F G H))) :
        tensorObj.{v, u} (𝒞 := inst_1) (tensorObj.{v, u} (𝒞 := inst_1) (Functor.obj lim F) (Functor.obj lim G))
            (Functor.obj lim H) ⟶
          Functor.obj lim (tensorObj (𝒞 := Functor.category) F (tensorObj (𝒞 := Functor.category) G H)))
      (limit.π (tensorObj (𝒞 := Functor.category) F (tensorObj (𝒞 := Functor.category) G H)) j))
    (CategoryStruct.comp (Y := limit (tensorObj (𝒞 := Functor.category) F (tensorObj (𝒞 := Functor.category) G H)))
      (Iso.hom (α_ (Functor.obj lim F) (Functor.obj lim G) (Functor.obj lim H)) ≫
        CategoryStruct.comp (Y :=
          tensorObj.{v, u} (𝒞 := inst_1) (Functor.obj lim F) (Functor.obj lim (tensorObj (𝒞 := Functor.category) G H)))
          (𝟙 (Functor.obj lim F) ⊗ₘ
            (fun (F G : Functor.{w, v, w, u} J C) ↦
                (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 := ⋯ } } :
                  Cone.pt.{w, v, w, u} (J := J) (F := 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 := ⋯ :
                            NatTrans
                              (Functor.obj (Functor.const J) (tensorObj.{v, u} (𝒞 := inst_1) (limit F) (limit G)))
                              (tensorObj (𝒞 := Functor.category) F G) } :
                        Cone (tensorObj (𝒞 := Functor.category) F G) } ⟶
                    limit (tensorObj (𝒞 := Functor.category) F G)))
              G H)
          ((fun (F G : Functor.{w, v, w, u} J C) ↦
              (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 := ⋯ } } :
                Cone.pt.{w, v, w, u} (J := J) (F := 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 := ⋯ :
                          NatTrans (Functor.obj (Functor.const J) (tensorObj.{v, u} (𝒞 := inst_1) (limit F) (limit G)))
                            (tensorObj (𝒞 := Functor.category) F G) } :
                      Cone (tensorObj (𝒞 := Functor.category) F G) } ⟶
                  limit (tensorObj (𝒞 := Functor.category) F G)))
            F (tensorObj (𝒞 := Functor.category) G H)))
      (limit.π (tensorObj (𝒞 := Functor.category) F (tensorObj (𝒞 := Functor.category) G H)) j))

Tactic:
dsimp