Corollary 9.1.7.7. Let $\kappa $ be a small regular cardinal and let $\operatorname{\mathcal{D}}$ be an $\infty $-category. If $\operatorname{\mathcal{D}}$ admits $\kappa $-small colimits and small $\kappa $-filtered colimits, then $\operatorname{\mathcal{D}}$ admits all small colimits. Moreover, if $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ is a functor of $\infty $-categories which preserves $\kappa $-small colimits and small $\kappa $-filtered colimits, then $G$ preserves all small colimits.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof. This is a special case of Corollary 9.1.7.6, since every small simplicial set $K$ can be realized as a (small) $\kappa $-filtered colimit of $\kappa $-small simplicial sets. For example, we can write $K$ as the union of all $\kappa $-small simplicial subsets of itself. $\square$