Remark 7.5.8.3. Let $\overline{\mathscr {F}}: \operatorname{\mathcal{C}}^{\triangleright } \rightarrow \operatorname{Set_{\Delta }}$ be a categorical colimit diagram of simplicial sets. Then $\overline{\mathscr {F}}$ is also a homotopy colimit diagram of simplicial sets, in the sense of Definition 7.5.7.3. This follows from the observation that every localization of simplicial sets is a weak homotopy equivalence (Remark 6.3.1.16).
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$