Notation 9.2.2.14. Let $\operatorname{\mathcal{QC}}$ denote the $\infty $-category of (small) $\infty $-categories (Construction 5.5.4.1). For every pair of regular cardinals $\kappa \leq \lambda $, we define a subcategory $\operatorname{\mathcal{QC}}^{ (\kappa ,\lambda )-\mathrm{ccomp} } \subseteq \operatorname{\mathcal{QC}}$ as follows:
An object $\operatorname{\mathcal{C}}$ of $\operatorname{\mathcal{QC}}$ belongs to $\operatorname{\mathcal{QC}}^{ (\kappa ,\lambda )-\mathrm{ccomp} }$ if and only if it is a $(\kappa ,\lambda )$-cocomplete $\infty $-category, in the sense of Definition 9.2.1.3.
A morphism $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ of $\operatorname{\mathcal{QC}}$ belongs to $\operatorname{\mathcal{QC}}^{ (\kappa ,\lambda )-\mathrm{ccomp} }$ if and only if it is a $(\kappa ,\lambda )$-finitary functor, in the sense of Definition 9.2.2.6.
More generally, if $\mu $ is any uncountable cardinal, we let $\operatorname{\mathcal{QC}}^{ (\kappa ,\lambda )-\mathrm{ccomp} }_{< \mu }$ denote the subcategory of $\operatorname{\mathcal{QC}}_{< \mu }$ whose objects are $\mu $-small $\infty $-categories which are $(\kappa ,\lambda )$-cocomplete, and whose morphisms are $(\kappa ,\lambda )$-finitary functors. In practice, we will be primarily interested in the case where $\mu $ is much larger than $\lambda $ (so that there are plenty of examples of such $\infty $-categories).