Kerodon

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Corollary 9.1.7.16. Let $\kappa $ and $\lambda $ be regular cardinals satisfying $\kappa \triangleleft \lambda $. Then every $\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 $\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 $\operatorname{\mathcal{QC}}$.