Warning 8.4.0.5. In ยง8.4.5, we will show that every $\infty $-category $\operatorname{\mathcal{C}}$ admits a cocompletion $\widehat{\operatorname{\mathcal{C}}}$ (Proposition 8.4.5.3). Beware that, if $\operatorname{\mathcal{C}}$ is not essentially small, then $\widehat{\operatorname{\mathcal{C}}}$ cannot necessarily be identified with the $\infty $-category $\operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}})$ (Warning 8.4.3.4). However, if $\operatorname{\mathcal{C}}$ is locally small, then it can be identified with a full subcategory of $\operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}})$ (see Construction 8.4.5.5).
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$