Free biholomorphic functions and operator model theory,II

This is a continuation of the paper entitled "Free biholomorphic functions and operator model theory", in our attempt to transfer the free analogue of Nagy-Foias theory from the unit ball $[B(\cH)^n]_1$ to other noncommutative domains and varieties in $B(\cH)^n$, using appropriate maps. The present paper treats the completely non-coisometric case and the case of noncommutative varieties.