Let $P$ be a small category, and let $F:P\to M$ be a diagram in a left-proper combinatorial model category $M$. We say that $F$ is a *diagram of cofibrations* if for every object $p\in P$, $F(p)$ is cofibrant, and for any $f:p\to p'$ in $\mathrm{Arr}(P)$, $F(f)$ is a cofibration.

Recall that a diagram $F: P\to M$ is called projectively cofibrant if the map $\emptyset \to F$ in $M^P$ has the left lifting property with respect to the class of morphisms that are objectwise trivial fibrations.

An easy example of when this holds is a for spans. Such diagrams are projectively cofibrant if and only if they are diagrams of cofibrations. Another example of such a diagram shape is the category of natural numbers (viewed as an ordered set).

Are there any known rules of thumb for when we can say that a diagram of cofibrations is projectively cofibrant?

If no general result exists, I'm specifically interested in the case where $M$ is the category of simplicial sets and $P$ is a poset.

Edit: Since the original question was answered in the comments, I'd like to add the conditions that the poset $P$ is finite and Thomason-contractible (that is, its nerve is a weakly contractible simplicial set).

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