  In this article we continue the development of a theory of noncommutative
motives. We construct categories of A1-homotopy noncommutative motives,
described their universal properties, and compute their spectra of morphisms in
terms of Karoubi-Villamayor's K-theory (KV) and Weibel's homotopy K-theory
(KH). As an application, we obtain a complete classification of all the natural
transformations defined on KV, KH. This leads to a streamlined construction of
Weibel's homotopy Chern character from KV to periodic cyclic homology. Along
the way we extend Dwyer-Friedlander's etale K-theory to the noncommutative
world, and develop the universal procedure of forcing a functor to preserve
filtered homotopy colimits.
