Definition 8.4.5.1. Let $\mathbb {K}$ be a collection of simplicial sets and let $h: \operatorname{\mathcal{C}}\rightarrow \widehat{\operatorname{\mathcal{C}}}$ be a functor of $\infty $-categories. We say that $h$ 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 $\operatorname{Fun}^{\mathbb {K}-\mathrm{cocont}}( \widehat{\operatorname{\mathcal{C}}}, \operatorname{\mathcal{D}}) \rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$.
Here $\operatorname{Fun}^{\mathbb {K}-\mathrm{cocont}}( \widehat{\operatorname{\mathcal{C}}}, \operatorname{\mathcal{D}})$ denotes the $\infty $-category of $\mathbb {K}$-cocontinuous functors from $\widehat{\operatorname{\mathcal{C}}}$ to $\operatorname{\mathcal{D}}$ (see Definition 7.6.6.22).