Definition 8.7.3.1. Let $\mathbb {K}$ be a collection of simplicial sets. We say that a functor $T: \operatorname{\mathcal{QC}}\rightarrow \operatorname{\mathcal{QC}}$ is a $\mathbb {K}$-cocompletion functor if there exists a natural transformation $\eta : \operatorname{id}_{\operatorname{\mathcal{QC}}} \rightarrow T$ satisfying the following conditions:
For every small $\infty $-category $\operatorname{\mathcal{C}}$, the functor $\eta _{\operatorname{\mathcal{C}}}: \operatorname{\mathcal{C}}\rightarrow T(\operatorname{\mathcal{C}})$ exhibits $T(\operatorname{\mathcal{C}})$ as a $\mathbb {K}$-cocompletion of $\operatorname{\mathcal{C}}$.
For every functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ between small $\infty $-categories, the functor $T(F): T(\operatorname{\mathcal{C}}) \rightarrow T(\operatorname{\mathcal{D}})$ is $\mathbb {K}$-cocontinuous.
If these conditions are satisfied, we say that $\eta $ exhibits $T$ as a $\mathbb {K}$-cocompletion functor.