# Reedy model structures on oplax limits

Suppose $R$ is a category and $F:R\to Cat$ is a functor (or pseudofunctor). The oplax limit of $F$ is the category whose objects consist of an object $x_r \in F(r)$ for all $r$ together with a morphism $x_s \to F(d)(x_{r})$ for all morphisms $d:r\to s$ in $R$, satisfying obvious compatibility conditions.

I have some vague memory of reading a paper in which, given a functor as above in which $R$ is a Reedy category, each category $F(r)$ is a model category, and probably some other conditions, a "Reedy-type" model structure on the oplax limit (or perhaps some related category) was constructed. However, I have been totally unable to find this paper again; the closest I can find is this paper which considers "injective-type" model structures on lax limits. Can anyone point me to the paper I am thinking of?

(I am not interested in seeing proofs or "it seems like this should work" arguments written out in the answers. I only want the reference.)

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How does having a morphism $x_r \to F(d)(x_s)$ work (typecheck)? Isn't $F(d)$ a functor from $F(r)$ to $F(s)$, while $x_s$ is an object in $F(s)$? –  Jason Gross Jul 30 '13 at 18:04
Thanks, fixed. (Sometimes people talk about contravariant functors instead of covariant ones.) –  Mike Shulman Jul 31 '13 at 19:02

but his constructions (Definitions 3.3 and 5.2) have a fixed category at each object of $R$ and only the model structures are allowed to vary.