Example 8.4.5.2. Let $\kappa $ be an uncountable regular cardinal, and let $\mathbb {K}$ denote the collection of all $\kappa $-small simplicial sets. Then an $\infty $-category $\operatorname{\mathcal{C}}$ is $\mathbb {K}$-cocomplete if and only if it is $\kappa $-cocomplete, in the sense of Variant 7.6.6.7. A functor of $\infty $-categories $h: \operatorname{\mathcal{C}}\rightarrow \widehat{\operatorname{\mathcal{C}}}$ exhibits $\widehat{\operatorname{\mathcal{C}}}$ as a $\mathbb {K}$-cocompletion of $\operatorname{\mathcal{C}}$ if and only if it exhibits $\widehat{\operatorname{\mathcal{C}}}$ as a $\kappa $-cocompletion of $\operatorname{\mathcal{C}}$, in the sense of Definition 8.4.3.2.
In particular, if $\mathbb {K}$ is the collection of all small simplicial sets, then an $\infty $-category $\operatorname{\mathcal{C}}$ is a $\mathbb {K}$-cocomplete if and only if it is cocomplete, and a functor $h: \operatorname{\mathcal{C}}\rightarrow \widehat{\operatorname{\mathcal{C}}}$ exhibits $\widehat{\operatorname{\mathcal{C}}}$ as a $\mathbb {K}$-cocompletion of $\operatorname{\mathcal{C}}$ if and only if it exhibits $\widehat{\operatorname{\mathcal{C}}}$ as a cocompletion of $\operatorname{\mathcal{C}}$.