Proposition 8.7.3.9. Let $\mathbb {K}$ be a collection of simplicial sets and let $\lambda $ be an uncountable cardinal. Assume that every $\lambda $-small $\infty $-category $\operatorname{\mathcal{C}}$ admits a $\lambda $-small $\mathbb {K}$-cocompletion. Then:
- $(1)$
The inclusion functor $\iota : \operatorname{\mathcal{QC}}_{<\lambda }^{\mathbb {K}-\mathrm{cocont}} \hookrightarrow \operatorname{\mathcal{QC}}_{< \lambda }$ admits a left adjoint.
- $(2)$
If $T: \operatorname{\mathcal{QC}}_{< \lambda } \rightarrow \operatorname{\mathcal{QC}}_{<\lambda }^{\mathbb {K}-\mathrm{cocont}}$ is a functor, then a natural transformation $\eta : \operatorname{id}_{\operatorname{\mathcal{QC}}_{< \lambda } } \rightarrow \iota \circ T$ is the unit of an adjunction between $T$ and $\iota $ (in the sense of Definition 6.2.1.1) if and only if it exhibits $T$ as a $\mathbb {K}$-cocompletion functor (in the sense of Definition 8.7.3.3).