-- Lean 4 proof step from Mathlib.CategoryTheory.Monoidal.Cartesian.CommMon_ (goal)
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : CartesianMonoidalCategory.{v, u} C
inst_2 : BraidedCategory.{v, u} C
X : C
F : Functor.{v, u_1, u, u_1 + 1} Cᵒᵖ CommMonCat
α :
  (F ⋙
        forget CommMonCat (FC := fun (x1 x2 : CommMonCat.{u_1}) ↦ (↑x1 : Type u_1) →* (↑x2 : Type u_1)) (CC :=
          CommMonCat.carrier)).RepresentableBy
    X
this_1 : MonObj X :=
  MonObj.ofRepresentableBy.{u_1, v, u} X
    (F ⋙
      forget₂ CommMonCat (FC := fun (x1 x2 : CommMonCat.{u_1}) ↦ (↑x1 : Type u_1) →* (↑x2 : Type u_1)) (CC :=
        CommMonCat.carrier) MonCat (FD := fun (x1 x2 : MonCat.{u_1}) ↦ (↑x1 : Type u_1) →* (↑x2 : Type u_1)) (CD :=
        MonCat.carrier))
    α
this :
  Eq.{v + 1} (α := tensorObj.{v, u} (𝒞 := inst) X X ⟶ X) (μ : tensorObj.{v, u} (𝒞 := inst) X X ⟶ X)
    (((Equiv.symm
              (α.homEquiv' (CD := CommMonCat.carrier) (FD := fun (x1 x2 : CommMonCat.{u_1}) ↦
                (↑x1 : Type u_1) →* (↑x2 : Type u_1))) :
            Equiv.{u_1 + 1, v + 1}
              (ToType (FC := fun (x1 x2 : CommMonCat.{u_1}) ↦ (↑x1 : Type u_1) →* (↑x2 : Type u_1)) (CC :=
                CommMonCat.carrier) (F.obj (op (tensorObj.{v, u} (𝒞 := inst) X X))))
              (tensorObj.{v, u} (𝒞 := inst) X X ⟶ X)) :
          (a : ToType (F.obj (op (X ⊗ X)))) → X ⊗ X ⟶ X)
        (HMul.hMul (α :=
          ToType (FC := fun (x1 x2 : CommMonCat.{u_1}) ↦ (↑x1 : Type u_1) →* (↑x2 : Type u_1)) (CC :=
            CommMonCat.carrier) (F.obj (op (tensorObj.{v, u} (𝒞 := inst) X X))))
          (β :=
          ToType (FC := fun (x1 x2 : CommMonCat.{u_1}) ↦ (↑x1 : Type u_1) →* (↑x2 : Type u_1)) (CC :=
            CommMonCat.carrier) (F.obj (op (tensorObj.{v, u} (𝒞 := inst) X X))))
          (((α.homEquiv' (CD := CommMonCat.carrier) (FD := fun (x1 x2 : CommMonCat.{u_1}) ↦
                    (↑x1 : Type u_1) →* (↑x2 : Type u_1)) :
                  Equiv.{v + 1, u_1 + 1} (tensorObj.{v, u} (𝒞 := inst) X X ⟶ X)
                    (ToType (FC := fun (x1 x2 : CommMonCat.{u_1}) ↦ (↑x1 : Type u_1) →* (↑x2 : Type u_1)) (CC :=
                      CommMonCat.carrier) (F.obj (op (tensorObj.{v, u} (𝒞 := inst) X X))))) :
                (a : X ⊗ X ⟶ X) → ToType (F.obj (op (X ⊗ X))))
              (fst X X) :
            ToType (FC := fun (x1 x2 : CommMonCat.{u_1}) ↦ (↑x1 : Type u_1) →* (↑x2 : Type u_1)) (CC :=
              CommMonCat.carrier) (F.obj (op (tensorObj.{v, u} (𝒞 := inst) X X))))
          (((α.homEquiv' (CD := CommMonCat.carrier) (FD := fun (x1 x2 : CommMonCat.{u_1}) ↦
                    (↑x1 : Type u_1) →* (↑x2 : Type u_1)) :
                  Equiv.{v + 1, u_1 + 1} (tensorObj.{v, u} (𝒞 := inst) X X ⟶ X)
                    (ToType (FC := fun (x1 x2 : CommMonCat.{u_1}) ↦ (↑x1 : Type u_1) →* (↑x2 : Type u_1)) (CC :=
                      CommMonCat.carrier) (F.obj (op (tensorObj.{v, u} (𝒞 := inst) X X))))) :
                (a : X ⊗ X ⟶ X) → ToType (F.obj (op (X ⊗ X))))
              (snd X X) :
            ToType (FC := fun (x1 x2 : CommMonCat.{u_1}) ↦ (↑x1 : Type u_1) →* (↑x2 : Type u_1)) (CC :=
              CommMonCat.carrier) (F.obj (op (tensorObj.{v, u} (𝒞 := inst) X X))))) :
      tensorObj.{v, u} (𝒞 := inst) X X ⟶ X)
⊢ autoParam (Eq.{v + 1} (α := tensorObj.{v, u} (𝒞 := inst) X X ⟶ X) (Iso.hom (β_ X X) ≫ μ) μ) mul_comm._autoParam
have mul_comm : autoParam
  (Eq.{v + 1} (α := tensorObj.{v, u} (𝒞 := inst) X X ⟶ X)
    (Iso.hom (β_ X X) ≫
      ((Equiv.symm
              (α.homEquiv' (CD := CommMonCat.carrier) (FD := fun (x1 x2 : CommMonCat.{u_1}) ↦
                (↑x1 : Type u_1) →* (↑x2 : Type u_1))) :
            Equiv.{u_1 + 1, v + 1}
              (ToType (FC := fun (x1 x2 : CommMonCat.{u_1}) ↦ (↑x1 : Type u_1) →* (↑x2 : Type u_1)) (CC :=
                CommMonCat.carrier) (F.obj (op (tensorObj.{v, u} (𝒞 := inst) X X))))
              (tensorObj.{v, u} (𝒞 := inst) X X ⟶ X)) :
          (a : ToType (F.obj (op (X ⊗ X)))) → X ⊗ X ⟶ X)
        (HMul.hMul (α :=
          ToType (FC := fun (x1 x2 : CommMonCat.{u_1}) ↦ (↑x1 : Type u_1) →* (↑x2 : Type u_1)) (CC :=
            CommMonCat.carrier) (F.obj (op (tensorObj.{v, u} (𝒞 := inst) X X))))
          (β :=
          ToType (FC := fun (x1 x2 : CommMonCat.{u_1}) ↦ (↑x1 : Type u_1) →* (↑x2 : Type u_1)) (CC :=
            CommMonCat.carrier) (F.obj (op (tensorObj.{v, u} (𝒞 := inst) X X))))
          (((α.homEquiv' (CD := CommMonCat.carrier) (FD := fun (x1 x2 : CommMonCat.{u_1}) ↦
                    (↑x1 : Type u_1) →* (↑x2 : Type u_1)) :
                  Equiv.{v + 1, u_1 + 1} (tensorObj.{v, u} (𝒞 := inst) X X ⟶ X)
                    (ToType (FC := fun (x1 x2 : CommMonCat.{u_1}) ↦ (↑x1 : Type u_1) →* (↑x2 : Type u_1)) (CC :=
                      CommMonCat.carrier) (F.obj (op (tensorObj.{v, u} (𝒞 := inst) X X))))) :
                (a : X ⊗ X ⟶ X) → ToType (F.obj (op (X ⊗ X))))
              (fst X X) :
            ToType (FC := fun (x1 x2 : CommMonCat.{u_1}) ↦ (↑x1 : Type u_1) →* (↑x2 : Type u_1)) (CC :=
              CommMonCat.carrier) (F.obj (op (tensorObj.{v, u} (𝒞 := inst) X X))))
          (((α.homEquiv' (CD := CommMonCat.carrier) (FD := fun (x1 x2 : CommMonCat.{u_1}) ↦
                    (↑x1 : Type u_1) →* (↑x2 : Type u_1)) :
                  Equiv.{v + 1, u_1 + 1} (tensorObj.{v, u} (𝒞 := inst) X X ⟶ X)
                    (ToType (FC := fun (x1 x2 : CommMonCat.{u_1}) ↦ (↑x1 : Type u_1) →* (↑x2 : Type u_1)) (CC :=
                      CommMonCat.carrier) (F.obj (op (tensorObj.{v, u} (𝒞 := inst) X X))))) :
                (a : X ⊗ X ⟶ X) → ToType (F.obj (op (X ⊗ X))))
              (snd X X) :
            ToType (FC := fun (x1 x2 : CommMonCat.{u_1}) ↦ (↑x1 : Type u_1) →* (↑x2 : Type u_1)) (CC :=
              CommMonCat.carrier) (F.obj (op (tensorObj.{v, u} (𝒞 := inst) X X))))))
    (((Equiv.symm
            (α.homEquiv' (CD := CommMonCat.carrier) (FD := fun (x1 x2 : CommMonCat.{u_1}) ↦
              (↑x1 : Type u_1) →* (↑x2 : Type u_1))) :
          Equiv.{u_1 + 1, v + 1}
            (ToType (FC := fun (x1 x2 : CommMonCat.{u_1}) ↦ (↑x1 : Type u_1) →* (↑x2 : Type u_1)) (CC :=
              CommMonCat.carrier) (F.obj (op (tensorObj.{v, u} (𝒞 := inst) X X))))
            (tensorObj.{v, u} (𝒞 := inst) X X ⟶ X)) :
        (a : ToType (F.obj (op (X ⊗ X)))) → X ⊗ X ⟶ X)
      (HMul.hMul (α :=
        ToType (FC := fun (x1 x2 : CommMonCat.{u_1}) ↦ (↑x1 : Type u_1) →* (↑x2 : Type u_1)) (CC := CommMonCat.carrier)
          (F.obj (op (tensorObj.{v, u} (𝒞 := inst) X X))))
        (β :=
        ToType (FC := fun (x1 x2 : CommMonCat.{u_1}) ↦ (↑x1 : Type u_1) →* (↑x2 : Type u_1)) (CC := CommMonCat.carrier)
          (F.obj (op (tensorObj.{v, u} (𝒞 := inst) X X))))
        (((α.homEquiv' (CD := CommMonCat.carrier) (FD := fun (x1 x2 : CommMonCat.{u_1}) ↦
                  (↑x1 : Type u_1) →* (↑x2 : Type u_1)) :
                Equiv.{v + 1, u_1 + 1} (tensorObj.{v, u} (𝒞 := inst) X X ⟶ X)
                  (ToType (FC := fun (x1 x2 : CommMonCat.{u_1}) ↦ (↑x1 : Type u_1) →* (↑x2 : Type u_1)) (CC :=
                    CommMonCat.carrier) (F.obj (op (tensorObj.{v, u} (𝒞 := inst) X X))))) :
              (a : X ⊗ X ⟶ X) → ToType (F.obj (op (X ⊗ X))))
            (fst X X) :
          ToType (FC := fun (x1 x2 : CommMonCat.{u_1}) ↦ (↑x1 : Type u_1) →* (↑x2 : Type u_1)) (CC :=
            CommMonCat.carrier) (F.obj (op (tensorObj.{v, u} (𝒞 := inst) X X))))
        (((α.homEquiv' (CD := CommMonCat.carrier) (FD := fun (x1 x2 : CommMonCat.{u_1}) ↦
                  (↑x1 : Type u_1) →* (↑x2 : Type u_1)) :
                Equiv.{v + 1, u_1 + 1} (tensorObj.{v, u} (𝒞 := inst) X X ⟶ X)
                  (ToType (FC := fun (x1 x2 : CommMonCat.{u_1}) ↦ (↑x1 : Type u_1) →* (↑x2 : Type u_1)) (CC :=
                    CommMonCat.carrier) (F.obj (op (tensorObj.{v, u} (𝒞 := inst) X X))))) :
              (a : X ⊗ X ⟶ X) → ToType (F.obj (op (X ⊗ X))))
            (snd X X) :
          ToType (FC := fun (x1 x2 : CommMonCat.{u_1}) ↦ (↑x1 : Type u_1) →* (↑x2 : Type u_1)) (CC :=
            CommMonCat.carrier) (F.obj (op (tensorObj.{v, u} (𝒞 := inst) X X)))))))
  mul_comm._autoParam
Tactic: simp_rw [this, ← α.homEquiv'.apply_eq_iff_eq, α.homEquiv'_comp,
    Equiv.apply_symm_apply, map_mul, ← α.homEquiv'_comp, op_tensorObj,
    braiding_hom_fst, braiding_hom_snd, _root_.mul_comm]