  We extend the Schoenberg correspondence for universal independences by
Sch\"urmann and Vo{\ss} to the multifaced, multistate setting of Manzel and
Sch\"urmann, covering, e.g., Voiculescu's bifreeness as well as Bo\.zejko and
Speicher's c-free independence. At the same time, we free the proof in the
single-face-single-state situation from its dependence on Muraki's
classification theorem.
