Definition 7.5.6.1. Let $\operatorname{\mathcal{C}}$ be a small category. We say that a diagram of simplicial sets $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$ is projectively cofibrant if, for every levelwise trivial Kan fibration $\beta : \mathscr {G}' \rightarrow \mathscr {G}$, the induced map
\[ \operatorname{Hom}_{ \operatorname{Fun}(\operatorname{\mathcal{C}},\operatorname{Set_{\Delta }}) }( \mathscr {F}, \mathscr {G}' ) \rightarrow \operatorname{Hom}_{ \operatorname{Fun}(\operatorname{\mathcal{C}},\operatorname{Set_{\Delta }}) }( \mathscr {F}, \mathscr {G}) \]
is surjective. That is, every natural transformation $\alpha : \mathscr {F} \rightarrow \mathscr {G}$ factors through $\beta $.