Functors between Reedy Model Categories of Diagrams

If D is a Reedy category and M is a model category, the category M of D-diagrams in M is a model category under the Reedy model category structure. If C → D is a Reedy functor between Reedy categories, then there is an induced functor of diagram categories M → M. Our main result is a characterization of the Reedy functors C → D that induce right or left Quillen functors M → M 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.