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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 categorystructure. If $C \to D$ is a Reedy functor between Reedy categories, then thereis an induced functor of diagram categories $M^{D} \to M^{C}$. Our main resultis a characterization of the Reedy functors $C \to D$ that induce right or leftQuillen functors $M^{D} \to M^{C}$ for every model category $M$. We apply theseresults to various situations, and in particular show that certain importantsubdiagrams of a fibrant multicosimplicial object are fibrant.

Functors between Reedy model categories of diagrams