Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$

Corollary 7.4.5.3. Let $\operatorname{\mathcal{C}}$ be a small simplicial set. Then any diagram $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}$ admits a limit in the $\infty $-category $\operatorname{\mathcal{S}}$, given by the $\infty $-category $\operatorname{Fun}_{ / \operatorname{\mathcal{C}}}( \operatorname{\mathcal{C}}, \int _{\operatorname{\mathcal{C}}} \mathscr {F} )$.

Proof. Apply Corollary 7.4.5.2 to the left fibration $U: \int _{\operatorname{\mathcal{C}}} \mathscr {F} \rightarrow \operatorname{\mathcal{C}}$ of Example 5.7.2.9. $\square$