Corollary 7.5.6.14. Let $\operatorname{\mathcal{C}}$ be a small category and let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$ be a projectively cofibrant diagram of simplicial sets. Then the comparison map $ \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F} ) \twoheadrightarrow \varinjlim ( \mathscr {F} )$ of Remark 5.3.2.9 is a weak homotopy equivalence.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof. By virtue of Remark 7.5.6.13, it will suffice to show that the natural transformation $\alpha : \mathscr {F}_{+} \twoheadrightarrow \mathscr {F}$ of Construction 7.5.6.8 induces a weak homotopy equivalence $\varinjlim (\alpha ): \varinjlim ( \mathscr {F}_{+} ) \rightarrow \varinjlim ( \mathscr {F} )$. This is a special case of Proposition 7.5.6.7, since $\alpha $ is a levelwise weak homotopy equivalence between projectively cofibrant diagrams (Proposition 7.5.6.9). $\square$