TITLE: About elegant Reedy categories
QUESTION [5 upvotes]: I discovered today the notion of elegant Reedy category introduced in the paper Reedy categories and the $\Theta$-construction of Julia E. Bergner and Charles Rezk. An interesting property of such categories is the following (quoting nlab elegant Reedy category):
if $R$ is an elegant Reedy category and $M$ is a model category in which the cofibrations are exactly the monomorphisms, then the Reedy model structure and the injective model structure on $M^{R^{op}}$ coincide.
This is stated in Proposition 3.15 of the paper of Bergner and Rezk for $M$ being a simplicial set-valued presheaf.
I was wondering if it would be possible to obtain a kind of dual statement of this, something like
"if $R$ is a "coelegant" Reedy category and $M$ is a model category in which the fibrations are exactly the epimorphisms, then the Reedy model structure and the projective model structure on $M^{R^{op}}$ coincide".
At first sight it seems possible to do so, but I may have missed something.

REPLY [5 votes]: Actually, I think the statement that you quoted from the nLab is wrong.  It was copied from v1 of the Bergner-Rezk paper, but v2 corrected the statement to be only about simplicial presheaves on $R$ (although I think the proof would probably work for presheaves on $R$ with values in any other presheaf category).  I've just fixed the nLab as well.