∀ (R : Type u_1) {M : Type u_2} {N : Type u_3} [inst : CommSemiring R] [inst_1 : Monoid M] [inst_2 : Monoid N]
  (f : M →* N),
  (Bialgebra.counitAlgHom R (MonoidAlgebra R N)).comp (MonoidAlgebra.mapDomainAlgHom R R f) =
    Bialgebra.counitAlgHom R (MonoidAlgebra R M)