∀ {R : Type u_1} {A : Type u_2} {C : Type u_3} [inst : CommSemiring R] [inst_1 : StarRing R]
  [inst_2 : NonUnitalSemiring A] [inst_3 : StarRing A] [inst_4 : Module R A] [inst_5 : SMulCommClass R A A]
  [inst_6 : IsScalarTower R A A] [inst_7 : Semiring C] [inst_8 : Algebra R C] [inst_9 : StarRing C] [StarModule R C]
  (φ : Unitization R A →⋆ₐ[R] C) (x : Unitization R A),
  (↑↑(Unitization.lift
                (φ.toNonUnitalStarAlgHom.comp
                    (Unitization.inrNonUnitalStarAlgHom R A)).toNonUnitalAlgHom).toRingHom).toFun
      (star x) =
    star
      ((↑↑(Unitization.lift
                  (φ.toNonUnitalStarAlgHom.comp
                      (Unitization.inrNonUnitalStarAlgHom R A)).toNonUnitalAlgHom).toRingHom).toFun
        x)