Schoenberg correspondence for multifaced independence

We extend the Schoenberg correspondence for universal independences by Sch"urmann & Vo{\ss} to the multivariate setting of Manzel & Sch"urmann, covering, e.g., Voiculescu’s bifreeness as well as Bo{.z}ejko & Speicher’s c-free independence. At the same time, we free the proof in the univariate situation from its dependence on Muraki’s ``5 Independences Theorem’'.