Definition 9.2.1.16. Let $\kappa \leq \lambda \leq \mu $ be regular cardinals, where $\mu $ is uncountable. We say that a functor $T: \operatorname{\mathcal{QC}}_{< \mu } \rightarrow \operatorname{\mathcal{QC}}_{< \mu }$ is an $\operatorname{Ind}_{\kappa }^{\lambda }$-completion functor if there exists a natural transformation $\eta : \operatorname{id}_{ \operatorname{\mathcal{QC}}_{< \mu } } \rightarrow T$ satisfying the following conditions:
For every $\mu $-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 an $\operatorname{Ind}_{\kappa }^{\lambda }$-completion of $\operatorname{\mathcal{C}}$.
For every functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ between $\mu $-small $\infty $-categories, the functor $T(F): T(\operatorname{\mathcal{C}}) \rightarrow T(\operatorname{\mathcal{D}})$ is $(\kappa ,\lambda )$-finitary.
If these conditions are satisfied, we say that $\eta $ exhibits $T$ as an $\operatorname{Ind}_{\kappa }^{\lambda }$-completion functor.