Definition 8.4.5.1. Let $\mathbb {K}$ be a collection of simplicial sets. We say that an $\infty $-category $\operatorname{\mathcal{C}}$ is $\mathbb {K}$-cocomplete if it admits $K$-indexed colimits, for each $K \in \mathbb {K}$. If $\operatorname{\mathcal{C}}$ is a $\mathbb {K}$-cocomplete $\infty $-category, we say that a functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is $\mathbb {K}$-cocontinuous if it preserves $K$-indexed colimits, for each $K \in \mathbb {K}$. In this case, we let $\operatorname{Fun}^{\mathbb {K}-\mathrm{cocont}}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$ denote the full subcategory of $\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$ spanned by the $\mathbb {K}$-cocontinuous functors from $\operatorname{\mathcal{C}}$ to $\operatorname{\mathcal{D}}$.
We say that 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 the following conditions are satisfied:
The $\infty $-category $\widehat{\operatorname{\mathcal{C}}}$ is $\mathbb {K}$-cocomplete.
For every $\mathbb {K}$-cocomplete $\infty $-category $\operatorname{\mathcal{D}}$, precomposition with $h$ induces an equivalence of $\infty $-categories $\operatorname{Fun}^{\mathbb {K}-\mathrm{cocont}}( \widehat{\operatorname{\mathcal{C}}}, \operatorname{\mathcal{D}}) \rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$.