Notation 9.2.3.4. Let $\operatorname{\mathcal{QC}}$ denote the $\infty $-category of (small) $\infty $-categories (Construction 5.5.4.1). For every regular cardinal $\kappa $, we define a subcategory $\operatorname{\mathcal{QC}}^{\kappa -\mathrm{seq}} \subseteq \operatorname{\mathcal{QC}}$ as follows:
An object $\operatorname{\mathcal{C}}$ of $\operatorname{\mathcal{QC}}$ belongs to $\operatorname{\mathcal{QC}}^{ \kappa -\mathrm{seq} }$ if and only if it is $\kappa $-sequentially cocomplete.
A morphism $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ of $\operatorname{\mathcal{QC}}$ belongs to $\operatorname{\mathcal{QC}}^{\kappa -\mathrm{seq}}$ if and only if it preserves $\kappa $-sequential colimits, in the sense of Definition 9.2.2.6.
More generally, if $\lambda $ is any uncountable cardinal, we let $\operatorname{\mathcal{QC}}^{ \kappa -\mathrm{seq} }_{< \lambda }$ denote the subcategory of $\operatorname{\mathcal{QC}}_{< \lambda }$ whose objects are $\lambda $-small $\infty $-categories which are $\kappa $-sequentially cocomplete, and whose morphisms are functors which preserve $\kappa $-seuqential colimits.