  If H is a commutative connected graded Hopf algebra over a commutative ring
k, then a certain canonical k-algebra homomorphism H -> H (x) QSym is defined,
where QSym denotes the Hopf algebra of quasisymmetric functions over k. This
homomorphism generalizes the "internal comultiplication" on QSym, and extends
what Hazewinkel (in Section 18.24 of his "Witt vectors") calls the Bernstein
homomorphism.
  We construct this homomorphism with the help of the universal property of
QSym as a combinatorial Hopf algebra (a well-known result by Aguiar, Bergeron
and Sottile) and extension of scalars (the commutativity of H allows us to
consider, for example, H (x) QSym as an H-Hopf algebra, and this change of
viewpoint significantly extends the reach of the universal property).
