∀ {R : Type u} {A : Type v} {B : Type w} {C : Type w'} [inst : CommSemiring R] [inst_1 : NonUnitalNonAssocSemiring A]
  [inst_2 : Module R A] [inst_3 : Star A] [inst_4 : NonUnitalNonAssocSemiring B] [inst_5 : Module R B] [inst_6 : Star B]
  [inst_7 : NonUnitalNonAssocSemiring C] [inst_8 : Module R C] [inst_9 : Star C] (S : NonUnitalStarSubalgebra R A)
  (g : B →⋆ₙₐ[R] C) (f : A →⋆ₙₐ[R] B),
  NonUnitalStarSubalgebra.map g (NonUnitalStarSubalgebra.map f S) = NonUnitalStarSubalgebra.map (g.comp f) S