Exercise 7.5.6.3 (Well-Founded Diagrams). Let $(Q, \leq )$ be a well-founded partially ordered set. Show that a diagram of simplicial sets $\mathscr {F}: Q \rightarrow \operatorname{Set_{\Delta }}$ is projectively cofibrant if and only if, for each element $q \in Q$, the associated map $\varinjlim _{ p < q} \mathscr {F}(p) \rightarrow \mathscr {F}(q)$ is a monomorphism of simplicial sets (compare with Proposition 4.5.6.6).
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$