$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Corollary 9.1.7.2. Let $\lambda $ be an uncountable regular cardinal. Then every $\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 $\lambda $-directed partially ordered set and each $\operatorname{\mathcal{C}}_{\alpha }$ is a $\lambda $-small $\infty $-category. Moreover, $\operatorname{\mathcal{C}}$ is also a colimit of the diagram $\{ \operatorname{\mathcal{C}}_{\alpha } \} _{\alpha \in A}$ in the $\infty $-category $\operatorname{\mathcal{QC}}$.
Proof.
Let $\{ \operatorname{\mathcal{C}}_{\alpha } \} _{\alpha \in A}$ be the collection of all $\lambda $-small simplicial subsets of $\operatorname{\mathcal{C}}$ which are $\infty $-categories. It follows from Proposition 9.1.7.1 that the index set $A$ is $\lambda $-directed (where we write $\alpha \leq \beta $ if $\operatorname{\mathcal{C}}_{\alpha }$ is contained in $\operatorname{\mathcal{C}}_{\beta }$), and that the canonical map $\varinjlim _{\alpha \in A} \operatorname{\mathcal{C}}_{\alpha } \rightarrow \operatorname{\mathcal{C}}$ is an isomorphism. The final assertion follows from Corollary 9.1.6.3.
$\square$