Warning 7.1.4.24. In the situation of Corollary 7.1.4.23, the inclusion functor $\operatorname{\mathcal{C}}' \hookrightarrow \operatorname{\mathcal{C}}$ generally does not preserve the colimit of the diagram $u$. If $C = \varinjlim (u)$ is a colimit of $u$ in the $\infty $-category $\operatorname{\mathcal{C}}$, then $C$ usually does not belong to $\operatorname{\mathcal{C}}'$. The colimit of $u$ in the $\infty $-category $\operatorname{\mathcal{C}}'$ is instead given by the localization $L(C)$.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$