∀ {R : Type u_4} {A : Type u_1} {B : Type u_2} {C : Type u_3} [inst : CommSemiring R] [inst_1 : Semiring A]
  [inst_2 : Algebra R A] [inst_3 : Star A] [inst_4 : Semiring B] [inst_5 : Algebra R B] [inst_6 : Star B]
  [inst_7 : Semiring C] [inst_8 : Algebra R C] [inst_9 : Star C],
  Function.LeftInverse (fun f => ((StarAlgHom.fst R B C).comp f, (StarAlgHom.snd R B C).comp f)) fun f => f.1.prod f.2