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

The Bernstein homomorphism via Aguiar-Bergeron-Sottile universality