Kerodon

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

Corollary 9.1.8.11. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category which is finite when viewed as a simplicial set. Then $\operatorname{\mathcal{C}}$ is filtered if and only if it has a final object.

Proof. This follows from Proposition 9.1.8.10 (applied in the special case $\kappa = \aleph _0$), since the $\infty $-category $\operatorname{\mathcal{C}}$ is automatically idempotent complete (Example 8.5.4.5). $\square$