$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Lemma 9.1.8.4. Let $\kappa $ be a regular cardinal and let $\operatorname{\mathcal{C}}$ be a simplicial set which is a $\kappa $-filtered $\infty $-category which satisfies condition $(\ast )$ of Lemma 9.1.8.2. Then $\operatorname{\mathcal{C}}$ can be realized as the colimit of a diagram
\[ A \rightarrow \operatorname{Set_{\Delta }}\quad \quad (\alpha \in A) \mapsto X_{\alpha }, \]
where $(A, \leq )$ is a $\kappa $-directed partially ordered set and each $X_{\alpha }$ is a simplicial set of the form $K_{\alpha }^{\triangleright }$ for some $\kappa $-small simplicial set $K_{\alpha }$. Moreover, if $\operatorname{\mathcal{C}}$ is $\mu $-small for some regular cardinal $\lambda $ with $\kappa \trianglelefteq \lambda $, then we can arrange that $A$ is also $\lambda $-small.
Proof.
Fix a regular cardinal $\lambda $ such that $\operatorname{\mathcal{C}}$ is $\lambda $-small. If $\kappa \trianglelefteq \lambda $, then we can choose a $\lambda $-small collection $\{ K_{\alpha } \} _{\alpha \in A}$ of $\kappa $-small simplicial subsets of $\operatorname{\mathcal{C}}$, such that every $\kappa $-small simplicial subset of $\operatorname{\mathcal{C}}$ is contained in some $K_{\alpha }$ (Lemma 9.1.7.18). Since $\operatorname{\mathcal{C}}$ satisfies condition $(\ast )$, each $K_{\alpha }$ is contained in a larger simplicial subset $X_{\alpha } \subseteq \operatorname{\mathcal{C}}$ which is isomorphic to the cone $K_{\alpha }^{\triangleright }$. We regard $A$ as a partially ordered set, where $\alpha \leq \beta $ if $X_{\alpha }$ is contained in $X_{\beta }$. Then $(A, \leq )$ is automatically $\kappa $-directed (Remark 9.1.7.19), and the colimit $\varinjlim _{\alpha \in A} X_{\alpha }$ identifies with the union $\bigcup _{\alpha } X_{\alpha } = \operatorname{\mathcal{C}}$.
$\square$