Functors between Reedy model categories of diagrams

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.