Notation 7.6.6.26. Let $\lambda $ be an uncountable cardinal, and let $\operatorname{\mathcal{QC}}_{< \lambda }$ denote the $\infty $-category of $\lambda $-small $\infty $-categories (Variant 5.5.3.11). For every collection of simplicial sets $\mathbb {K}$, we define a subcategory $\operatorname{\mathcal{QC}}_{< \lambda }^{\mathbb {K}-\mathrm{comp}} \subseteq \operatorname{\mathcal{QC}}_{< \lambda }$ as follows:
An object $\operatorname{\mathcal{C}}\in \operatorname{\mathcal{QC}}_{< \lambda }$ belongs to the subcategory $\operatorname{\mathcal{QC}}^{\mathbb {K}-\mathrm{comp}}_{< \lambda }$ if and only if $\operatorname{\mathcal{C}}$ is $\mathbb {K}$-complete.
A morphism $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ of $\operatorname{\mathcal{QC}}_{< \lambda }$ belongs to the subcategory $\operatorname{\mathcal{QC}}^{\mathbb {K}-\mathrm{comp}}_{< \lambda }$ if and only if it is $\mathbb {K}$-continuous.
Similarly, we let $\operatorname{\mathcal{QC}}_{< \lambda }^{\mathbb {K}-\mathrm{ccomp}} \subseteq \operatorname{\mathcal{QC}}_{< \lambda }$ denote the subcategory whose objects are $\mathbb {K}$-cocomplete $\infty $-categories and whose morphisms are $\mathbb {K}$-cocontinuous functors.
We will sometimes be 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 $-categories $\operatorname{\mathcal{QC}}_{< \lambda }^{\mathbb {K}-\mathrm{comp}}$ and $\operatorname{\mathcal{QC}}_{< \lambda }^{\mathbb {K}-\mathrm{ccomp}}$ by $\operatorname{\mathcal{QC}}^{\kappa -\mathrm{comp}}_{< \lambda }$ and $\operatorname{\mathcal{QC}}^{\kappa -\mathrm{ccomp}}_{< \lambda }$, respectively. In the case where $\mathbb {K}$ is the collection of all small simplicial sets, we denote these by $\operatorname{\mathcal{QC}}_{< \lambda }^{\operatorname{\mathrm{comp}}}$ and $\operatorname{\mathcal{QC}}_{< \lambda }^{\operatorname{\mathrm{ccomp}}}$, respectively.