Let $M$ be a model category and $S \subseteq \operatorname{Mor}(M)$ a set of arrows in (the underlying strict category of) $M$. Recall that the left Bousfield localization $L_SM$ of $M$ with respect to $S$ is the model category structure on the same underlying category as $M$, with the same cofibrations, more weak equivalences, and correspondingly fewer fibrations; it is the universal model category structure with these properties in which the elements of $S$ are weak equivalences.

Let $M^\Delta$ denote the category of simplicial objects in $M$. It has two related model category structures: the *injective* one, in which cofibrations and weak equivalences are "levelwise" (and hence fibrations are complicated), and the *projective* one, in which fibrations and weak equivalences are levelwise (and cofibrations are more complicated). I am interested in both structures, so in some sense this question is really two questions.

Question:Is $(L_SM)^\Delta$ a left Bousfield localization of $M^\Delta$? What is an explicit description of the maps at which you localize?

I expect the answer is "yes", and that you can localize at the set of maps in $M^\Delta$ that are levelwise in $S$. These seems like the type of thing that should be provable by definition unpacking. But I got stuck somewhere, suggesting that maybe there is a subtlety. The worry is that perhaps there just aren't a lot of levelwise-$S$-maps in $M^\Delta$, so that testing just against the levelwise-$S$-maps might give false positives when you are trying to find out if an object is levelwise-$S$-local.