Proposition 7.5.6.9. Let $\operatorname{\mathcal{C}}$ be a small category and let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$ be a diagram of simplicial sets. Then the diagram $\mathscr {F}_{+}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$ of Construction 7.5.6.8 is projectively cofibrant, and the natural transformation $\alpha : \mathscr {F}_{+} \rightarrow \mathscr {F}$ is a levelwise weak homotopy equivalence. Moreover, $\alpha $ is also an epimorphism.
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Proof. Example 7.5.6.2 shows that the diagram $\mathscr {F}_{+}$ is projectively cofibrant and Remark 5.3.2.9 shows that $\alpha $ is an epimorphism. To complete the proof, it will suffice to show that for each object $C \in \operatorname{\mathcal{C}}$, the map $\alpha _{C}: \mathscr {F}_{+}(C) \rightarrow \mathscr {F}(C)$ is a weak homotopy equivalence of simplicial sets. Replacing $\operatorname{\mathcal{C}}$ by the slice category $\operatorname{\mathcal{C}}_{/C}$, we can reduce to the case where $C$ is a final object of $\operatorname{\mathcal{C}}$; in this case, we wish to prove that the comparison map
\[ \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F} ) \rightarrow \varinjlim ( \mathscr {F} ) \simeq \mathscr {F}(C) \]
is a weak homotopy equivalence. Note that this map admits a section, given by the inclusion map
\[ \iota : \mathscr {F}(C) \simeq \{ C\} \times _{ \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) } \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F} ) \rightarrow \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F} ). \]
We complete the proof by that our assumption that $C \in \operatorname{\mathcal{C}}$ is a final object guarantees that $\iota $ is right anodyne (Example 7.2.3.11). $\square$