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](http://arxiv.org/abs/1010.0717) 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.)



Reedy model structures on oplax limits
I think the paper I was thinking of when I asked this question was probably


* Markus Spitzweck, [Homotopy limits of model categories over inverse index categories](https://doi.org/10.1016/j.jpaa.2009.08.001), JPAA 214:6 (2010)


although he only considers the case of inverse categories rather than more general Reedy ones. A more recent reference that does deal with the case of general Reedy categories is


* Edouard Balzin, Reedy Model Structures in Families, [arXiv:1803.00681](https://arxiv.org/abs/1803.00681)


I have collected some more remarks and references on model structures of this sort at the nLab page [model structure on sections](https://ncatlab.org/nlab/show/model+structure+on+sections).


