If $\mathcal{M}$ is a cofibrantly generated model category and $\mathcal{C}$ is a small category, then we can give $\mathcal{M}^\mathcal{C}$ the projective model structure, in which weak equivalences and fibrations are transformations that are such on each object of $\mathcal{C}$. I recall seeing a generalization in which we take a subset $D$ of objects of $\mathcal{C}$ and say that a transformation $f$ is a weak equivalence or fibration if $f(d)$ is one *for every $d\in D$* (and require nothing at the other objects). Am I remembering correctly that this works in this generality, and does anyone have a reference for where this is written down? I'd like to use this but see no need to reinvent the wheel if I can just refer to it.



Looking for generalization of projective model structure
This model structure has appeared explicitly in the paper by Paul Balmer and Michel Matthey "[Codescent theory I: Foundations](https://www.sciencedirect.com/science/article/pii/S0166864104001828?via%3Dihub)". Theorem 3.5 establishes the existence of, what the authors called, the relative model structure.


But the ideas behind this model structure go back to Dwyer-Kan concept of an *orbit* introduced in the paper "[Singular functors and realization functors](https://www.sciencedirect.com/science/article/pii/1385725884900167)". If you choose the set of orbits to be the representable functors $hom\_{\mathcal{C}}(d,-)$ for all $d\in \mathcal{D}$, then this set of orbits defines the same $\mathcal{D}$-relative model structure on the category of $\mathcal{C}$-indexed diagrams.


