Definition 8.7.3.3. Let $\mathbb {K}$ be a collection of simplicial sets and let $\lambda $ be an uncountable cardinal. We say that a functor $T: \operatorname{\mathcal{QC}}_{< \lambda } \rightarrow \operatorname{\mathcal{QC}}_{< \lambda }$ 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 $\lambda $-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 $\lambda $-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.
In the special case where $\mathbb {K}$ is the collection of all small simplicial sets, we say that $T$ is a cocompletion functor if it is $\mathbb {K}$-cocompletion functor. More generally, if $\mathbb {K}$ is the collection of all $\kappa $-small simplicial sets (for some infinite cardinal $\kappa $), we say that $T$ is a $\kappa $-cocompletion functor if it is a $\mathbb {K}$-cocompletion functor.