This follows by combining point 3. of the theorem above with the fact that the matching categories satisfy the “Leibniz rule”, see Riehl, Verity, Observation 4.2.

This is proven in Hirschhorn, Proposition 15.10.4 in the case of $R = \Delta$, but the argument given there applies to any elegant Reedy category.

Examples

The simplex category$\Delta$ is a Reedy category with fibrant constants.

More generally, Joyal’s disk categories$\Theta_n$ are Reedy categories with fibrant constants.

Every direct category is a Reedy category with fibrant constants.

References

Philip Hirschhorn, Model categories and their localizations, volume 99 of Mathematical Surveys and Monographs, American Mathematical Society, 2009,