  This paper is an attempt to find large classes of noncommutative
multivariable functions $g:\Omega\subset [B(\cH)^n]_1^-\to B(\cH)^n$ for which
a reasonable operator model theory and dilation theory can be developed for the
noncommutative domain $g(\Omega)$. In other words, we want to transfer the free
analogue of Nagy-Foias theory from the unit ball $[B(\cH)^n]_1$ to other
noncommutative domains, using appropriate maps.
