Corollary 8.7.3.10. Let $\mathbb {K}$ be a collection of simplicial sets and let $\lambda $ be an uncountable cardinal. The following conditions are equivalent:
- $(1)$
Every $\lambda $-small $\infty $-category $\operatorname{\mathcal{C}}$ admits a $\mathbb {K}$-cocompletion which is also $\lambda $-small.
- $(2)$
There exists a $\mathbb {K}$-cocompletion functor $T: \operatorname{\mathcal{QC}}_{< \lambda } \rightarrow \operatorname{\mathcal{QC}}_{<\lambda }$.
Moreover, if these conditions are satisfied, then a functor $T: \operatorname{\mathcal{QC}}_{< \lambda } \rightarrow \operatorname{\mathcal{QC}}_{< \lambda }$ is a $\mathbb {K}$-cocompletion functor if and only if it factors through the subcategory $\operatorname{\mathcal{QC}}_{< \lambda }^{\mathbb {K}-\mathrm{cocont}}$ and is left adjoint to the inclusion functor $\operatorname{\mathcal{QC}}_{< \lambda }^{\mathbb {K}-\mathrm{cocont}} \hookrightarrow \operatorname{\mathcal{QC}}_{< \lambda }$.