Notation 8.7.3.7. Let $\lambda $ be an uncountable cardinal, and let $\operatorname{\mathcal{QC}}_{< \lambda }$ denote the $\infty $-category of $\lambda $-small $\infty $-categories (Variant 5.5.4.11). For every collection of simplicial sets $\mathbb {K}$, we define a subcategory $\operatorname{\mathcal{QC}}_{< \lambda }^{\mathbb {K}-\mathrm{cocont}} \subseteq \operatorname{\mathcal{QC}}_{< \lambda }$ as follows:
An object $\operatorname{\mathcal{C}}\in \operatorname{\mathcal{QC}}_{< \lambda }$ belongs to the subcategory $\operatorname{\mathcal{QC}}^{\mathbb {K}-\mathrm{cocont}}_{< \lambda }$ if and only if $\operatorname{\mathcal{C}}$ is $\mathbb {K}$-cocomplete: that is, it admits $K$-indexed colimits for each $K \in \mathbb {K}$
A morphism $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ of $\operatorname{\mathcal{QC}}_{< \lambda }$ belongs to the subcategory $\operatorname{\mathcal{QC}}^{\mathbb {K}-\mathrm{cocont}}_{< \lambda }$ if and only if it is $\mathbb {K}$-cocontinuous: that is, it preserves $K$-indexed colimits for each $K \in \mathbb {K}$.
We will be particularly interested in the special case where $\mathbb {K}$ is the collection of all $\kappa $-small simplicial sets, for some infinite cardinal $\kappa $. In this case, we denote the $\infty $-category $\operatorname{\mathcal{QC}}^{\mathbb {K}-\mathrm{cocont}}_{< \lambda }$ by $\operatorname{\mathcal{QC}}^{\kappa -\mathrm{cocont}}_{< \lambda }$.