Kerodon

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

Corollary 9.2.3.8. An $\infty $-category $\operatorname{\mathcal{C}}$ is $(\aleph _0, \aleph _1)$-cocomplete (in the sense of Definition 9.2.1.3) if and only if it is sequentially cocomplete (in the sense of Definition 7.6.5.1). If these conditions are satisfied, then a functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is $(\aleph _0, \aleph _1)$-finitary if and only if it preserves sequential colimits.

Proof. Apply Proposition 9.2.3.6 (and Remark 9.2.3.7) in the special case $\kappa = \aleph _0$ (see Example 9.2.3.2). $\square$