-- 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 : Functor.{w, v, w, u} J C
j : J
⊢ Eq (α := tensorObj.{v, u} (𝒞 := inst_1) (limit F) (tensorUnit.{v, u} C (𝒞 := inst_1)) ⟶ F.obj j)
    (Iso.hom (ρ_ (limit F)) ≫ limit.π F j)
    (CategoryStruct.comp (Y := limit F)
      (CategoryStruct.comp (Y :=
        tensorObj.{v, u} (𝒞 := inst_1) (limit F)
          (limit.{w, w, v, u} (J := J) ((𝟙_ (Functor.{w, v, w, u} J C)) (𝒞 := Functor.category))))
        (𝟙 (limit F) ⊗ₘ
            limit.lift (Functor.obj (Functor.const J) (tensorUnit.{v, u} C (𝒞 := inst_1)))
              { pt := tensorUnit.{v, u} C (𝒞 := inst_1),
                π :=
                  { app := fun (x : J) ↦ 𝟙 (tensorUnit.{v, u} C (𝒞 := inst_1)), naturality := ⋯ :
                    NatTrans.{w, v, w, u} (Functor.obj (Functor.const J) (tensorUnit.{v, u} C (𝒞 := inst_1)))
                      (Functor.obj (Functor.const J) (tensorUnit.{v, u} C (𝒞 := inst_1))) } } :
          tensorObj.{v, u} (𝒞 := inst_1) (limit F) (tensorUnit.{v, u} C (𝒞 := inst_1)) ⟶
            tensorObj.{v, u} (𝒞 := inst_1) (limit F)
              (limit.{w, w, v, u} (J := J) ((𝟙_ (Functor.{w, v, w, u} J C)) (𝒞 := Functor.category))))
        (CategoryStruct.comp (Y :=
            limit (tensorObj (𝒞 := Functor.category) F ((𝟙_ (Functor.{w, v, w, u} J C)) (𝒞 := Functor.category))))
            (limit.lift (tensorObj (𝒞 := Functor.category) F ((𝟙_ (Functor.{w, v, w, u} J C)) (𝒞 := Functor.category)))
              {
                pt :=
                  tensorObj.{v, u} (𝒞 := inst_1) (limit F)
                    (limit.{w, w, v, u} (J := J) ((𝟙_ (Functor.{w, v, w, u} J C)) (𝒞 := Functor.category))),
                π :=
                  {
                    app := fun (j : J) ↦
                      limit.π F j ⊗ₘ limit.π ((𝟙_ (Functor.{w, v, w, u} J C)) (𝒞 := Functor.category)) j,
                    naturality := ⋯ } })
            (limMap (Iso.hom (ρ_ F))) :
          tensorObj.{v, u} (𝒞 := inst_1) (limit F)
              (limit.{w, w, v, u} (J := J) ((𝟙_ (Functor.{w, v, w, u} J C)) (𝒞 := Functor.category))) ⟶
            limit F))
      (limit.π F j))
have simp : Eq (α := tensorObj.{v, u} (𝒞 := inst_1) (limit F) (tensorUnit.{v, u} C (𝒞 := inst_1)) ⟶ F.obj j)
  (Iso.hom (ρ_ (limit F)) ≫ limit.π F j)
  (limit F ◁
      limit.lift (Functor.obj (Functor.const J) (tensorUnit.{v, u} C (𝒞 := inst_1)))
        { pt := tensorUnit.{v, u} C (𝒞 := inst_1),
          π :=
            { app := fun (x : J) ↦ 𝟙 (tensorUnit.{v, u} C (𝒞 := inst_1)), naturality := sorry :
              NatTrans.{w, v, w, u} (Functor.obj (Functor.const J) (tensorUnit.{v, u} C (𝒞 := inst_1)))
                (Functor.obj (Functor.const J) (tensorUnit.{v, u} C (𝒞 := inst_1))) } } ≫
    (Functor.obj (C :=
            Cone (tensorObj (𝒞 := Functor.category) F ((𝟙_ (Functor.{w, v, w, u} J C)) (𝒞 := Functor.category))))
            (Cone.postcompose (Iso.hom (ρ_ F)))
            {
              pt :=
                tensorObj.{v, u} (𝒞 := inst_1) (limit F)
                  (limit.{w, w, v, u} (J := J) ((𝟙_ (Functor.{w, v, w, u} J C)) (𝒞 := Functor.category))),
              π :=
                {
                  app := fun (j : J) ↦
                    limit.π F j ⊗ₘ limit.π ((𝟙_ (Functor.{w, v, w, u} J C)) (𝒞 := Functor.category)) j,
                  naturality := sorry :
                  NatTrans
                    (Functor.obj (Functor.const J)
                      (tensorObj.{v, u} (𝒞 := inst_1) (limit F)
                        (limit.{w, w, v, u} (J := J) ((𝟙_ (Functor.{w, v, w, u} J C)) (𝒞 := Functor.category)))))
                    (tensorObj (𝒞 := Functor.category) F ((𝟙_ (Functor.{w, v, w, u} J C)) (𝒞 := Functor.category))) } :
              Cone
                (tensorObj (𝒞 := Functor.category) F ((𝟙_ (Functor.{w, v, w, u} J C)) (𝒞 := Functor.category))) }).π.app
      j)
Tactic: simp only [id_tensorHom, limit.lift_map, Category.assoc, limit.lift_π]