$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proposition 9.1.7.20. Let $\kappa $, $\lambda $, and $\mu $ be regular cardinals satisfying $\kappa \triangleleft \lambda $ and $\lambda \trianglelefteq \mu $. Then every $\mu $-small $\kappa $-filtered $\infty $-category $\operatorname{\mathcal{C}}$ can be realized as the colimit of some diagram
\[ A \rightarrow \operatorname{Set_{\Delta }}\quad \quad (\alpha \in A) \mapsto \operatorname{\mathcal{C}}_{\alpha }, \]
where $(A, \leq )$ is a $\mu $-small $\lambda $-directed partially ordered set and each $\operatorname{\mathcal{C}}_{\alpha }$ is a $\lambda $-small $\kappa $-filtered $\infty $-category. Moreover, $\operatorname{\mathcal{C}}$ is also a colimit of the diagram $\{ \operatorname{\mathcal{C}}_{\alpha } \} _{\alpha \in A}$ in the $\infty $-category $\operatorname{\mathcal{QC}}_{< \mu }$.
Proof.
Using Lemma 9.1.7.18, we can choose a $\mu $-small collection $\{ \operatorname{\mathcal{C}}_{\alpha } \} _{\alpha \in A}$ of $\lambda $-small simplicial subsets of $\operatorname{\mathcal{C}}$ such that every $\lambda $-small simplicial subset of $\operatorname{\mathcal{C}}$ is contained in some $\operatorname{\mathcal{C}}_{\alpha }$. Enlarging the simplicial subsets $\operatorname{\mathcal{C}}_{\alpha }$ if necessary, we may assume that they are $\kappa $-filtered $\infty $-categories (Proposition 9.1.7.14). The index set $A$ is then $\lambda $-directed by inclusion (Remark 9.1.7.19), and $\operatorname{\mathcal{C}}= \bigcup _{\alpha } \operatorname{\mathcal{C}}_{\alpha }$ is the colimit of the diagram $\{ \operatorname{\mathcal{C}}_{\alpha } \} $ in the category of simplicial sets. The final assertion follows from Corollary 9.1.6.3.
$\square$