$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proposition 9.1.8.5. Let $\kappa $ and $\lambda $ be regular cardinals satisfying $\kappa \triangleleft \lambda $, and let $\operatorname{\mathcal{C}}$ be an $\infty $-category which is $\kappa $-filtered and essentially $\lambda $-small. Then there exists a $\lambda $-small, $\kappa $-directed partially ordered set $A$ and a diagram
\[ \operatorname{N}_{\bullet }(A) \rightarrow \operatorname{\mathcal{QC}}_{< \lambda } \quad \quad (\alpha \in A) \mapsto \operatorname{\mathcal{E}}_{\alpha } \]
having colimit $\operatorname{\mathcal{C}}$ which satisfies the following condition:
- $(\ast )$
For each $\alpha \in A$, there exists a $\kappa $-small simplicial set $K_{\alpha }$ and a categorical equivalence $K^{\triangleright }_{\alpha } \rightarrow \operatorname{\mathcal{E}}_{\alpha }$.
Proof of Proposition 9.1.8.5.
Replacing $\operatorname{\mathcal{C}}$ by an equivalent $\infty $-category, we may assume that it is $\lambda $-small and satisfies condition $(\ast )$ of Lemma 9.1.8.2. Using Lemma 9.1.8.4, we can realize $\operatorname{\mathcal{C}}$ as the colimit of a diagram
\[ \mathscr {F}: A \rightarrow \operatorname{Set_{\Delta }}\quad \quad (\alpha \in A) \mapsto X_{\alpha }, \]
where each $X_{\alpha }$ has the form $K_{\alpha }^{\triangleright }$ for some $\kappa $-small simplicial set $K_{\alpha }$. Using Proposition 4.1.3.2, we can choose a levelwise categorical equivalence $\mathscr {F} \rightarrow \mathscr {F}'$ for some functor
\[ \mathscr {F}': A \rightarrow \operatorname{Set_{\Delta }}\quad \quad (\alpha \in A) \mapsto \operatorname{\mathcal{E}}_{\alpha } \]
carrying each $\alpha \in A$ to an $\infty $-category $\operatorname{\mathcal{E}}_{\alpha }$. By construction, each of the $\infty $-categories $\operatorname{\mathcal{E}}_{\alpha }$ essentially $\lambda $-small, so that $\mathscr {F}'$ determines a diagram $\operatorname{N}_{\bullet }(A) \rightarrow \operatorname{\mathcal{QC}}_{< \lambda }$. It follows from Corollary 9.1.6.3 that $\operatorname{\mathcal{C}}$ is a colimit of this diagram.
$\square$