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](http://ncatlab.org/nlab/show/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.



About elegant Reedy categories
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.


