Example 7.5.6.2. Let $\operatorname{\mathcal{C}}$ be a category and let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ be a morphism of simplicial sets. Then the diagram
is projectively cofibrant, in the sense of Definition 7.5.6.1. To prove this, we must show that for every levelwise trivial Kan fibration $\mathscr {G}' \rightarrow \mathscr {G}$ between functors $\mathscr {G}', \mathscr {G}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$, the induced map
is surjective. Using Proposition 5.3.3.24, we can identify $\theta $ with a pullback of the map $\operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \operatorname{\mathcal{E}}, \operatorname{N}_{\bullet }^{\mathscr {G}'}(\operatorname{\mathcal{C}}) ) \rightarrow \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \operatorname{\mathcal{E}}, \operatorname{N}_{\bullet }^{\mathscr {G} }(\operatorname{\mathcal{C}}) )$, which is surjective by virtue of Exercise 5.3.3.11.