2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/67760A duality transform for the coalgebra of the free difference quotient derivation-multiplication of an operator with respect to a free algebra of scalars is constructed. The dual object is realized in an algebra of matricial analytic functions endowed with yet another generalization of the difference quotient derivation.27 pages; key words and phrases: free difference quotient, derivation-comultiplication, generalized matricial resolvent, coalgebra duality, freeOperator AlgebrasQuantum AlgebraPrimary 16W30, 46L54; Secondary 47C05, 30E99, 30G30text