  Stable derivators provide an enhancement of triangulated categories as is
indicated by the existence of canonical triangulations. In this paper we show
that exact morphisms of stable derivators induce exact functors of canonical
triangulations, and similarly for arbitrary natural transformations. This
2-categorical refinement also provides a uniqueness statement concerning
canonical triangulations.
  These results rely on a more careful study of morphisms of derivators and
this study is of independent interest. We analyze the interaction of morphisms
of derivators with limits, colimits, and Kan extensions, including a discussion
of invariance and closure properties of the class of Kan extensions preserved
by a fixed morphism.
