  The Sz.-Nagy Foias characteristic function for a contraction has had a
rejuvenation in recent times due to a number of authors. Such a classical
object relates to an object of very contemporary interest, viz., the complete
Nevanlinna-Pick kernels. Indeed, a unitarily invariant kernel on the unit ball
{\em admits} a characteristic function if and only if it is a complete
Nevanlinna-Pick kernel. However, what has captured our curiosity are the recent
advancements in constructing characteristic functions for kernels that do not
have complete Nevanlinna-Pick property. In such cases, the reproducing kernel
Hilbert space which has served as the domain of the multiplication operator has
always been the vector-valued Drury-Arveson space (thus the Hardy space in case
of the unit disc).
  We present a unified framework for deriving characteristic functions for
kernels that allow a complete Nevanlinna-Pick factor. Notably, our approach not
only encapsulates all previously documented cases but also achieves a
remarkable level of generalization, thereby expanding the concept of the
characteristic function substantially. We also provide an explanation for the
prominence of the Drury-Arveson kernel in all previously established results by
showing that the Drury-Arveson kernel was the natural choice inherently
suitable for those situations.
