2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/172496We extend Voiculescu's approach to analytic subordination through the coalgebra of the free difference quotient to non-coassociative derivation-comultiplications appearing in free probability theory. We obtain new proofs of Voiculescu's analytic subordination results for freely Markovian triples, and for multiplication of unitaries which are free with amalgamation.17 pagesOperator AlgebrasQuantum Algebra46L54; 16W30text