  If $D$ is a Reedy category and $M$ is a model category, the category $M^{D}$
of $D$-diagrams in $M$ is a model category under the Reedy model category
structure. If $C \to D$ is a Reedy functor between Reedy categories, then there
is an induced functor of diagram categories $M^{D} \to M^{C}$. Our main result
is a characterization of the Reedy functors $C \to D$ that induce right or left
Quillen functors $M^{D} \to M^{C}$ for every model category $M$. We apply these
results to various situations, and in particular show that certain important
subdiagrams of a fibrant multicosimplicial object are fibrant.
