Abstract The goal of this article is to study the coaugmented curved A ∞ -coalgebra structure of the Koszul codual of a filtered dg algebra over a field k . More precisely, we first extend one result of B. Keller that allowed to compute the A ∞ -coalgebra structure of the Koszul codual of a nonnegatively graded connected algebra to the case of any unitary dg algebra provided with a nonnegative increasing filtration whose zeroth term is k . We then show how to compute the coaugmented curved A ∞ -coalgebra structure of the Koszul codual of a Poincaré-Birkhoff-Witt (PBW) deformation of an N -Koszul algebra.