Kerodon

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

Theorem 9.3.7.1. Let $\operatorname{\mathcal{C}}$ be a small $\infty $-category. Then $\operatorname{\mathcal{C}}$ is filtered if and only if the $\operatorname{Ind}$-completion $\operatorname{Ind}(\operatorname{\mathcal{C}})$ has a final object.

Proof of Theorem 9.3.7.1. Apply Corollary 9.3.7.5 in the special case $\kappa = \aleph _0$. $\square$