Definition 8.4.3.1. Let $\kappa $ be an uncountable regular cardinal. We say that a functor of $\infty $-categories $h: \operatorname{\mathcal{C}}\rightarrow \widehat{\operatorname{\mathcal{C}}}$ exhibits $\widehat{\operatorname{\mathcal{C}}}$ as a $\kappa $-cocompletion of $\operatorname{\mathcal{C}}$ if the following conditions are satisfied:
- $(1)$
The $\infty $-category $\widehat{\operatorname{\mathcal{C}}}$ is $\kappa $-cocomplete.
- $(2)$
For every $\kappa $-cocomplete $\infty $-category $\operatorname{\mathcal{D}}$, precomposition with $h$ induces an equivalence of $\infty $-categories $\operatorname{Fun}^{\kappa -\mathrm{cocont}}( \widehat{\operatorname{\mathcal{C}}}, \operatorname{\mathcal{D}}) \rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$.
Here $\operatorname{Fun}^{\kappa -\mathrm{cocont}}( \widehat{\operatorname{\mathcal{C}}}, \operatorname{\mathcal{D}})$ denotes the $\infty $-category of $\kappa $-cocontinuous functors from $\widehat{\operatorname{\mathcal{C}}}$ to $\operatorname{\mathcal{D}}$ (Notation 7.6.6.7).