Definition 8.4.3.2. 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 $\operatorname{Fun}^{\kappa }( \widehat{\operatorname{\mathcal{C}}}, \operatorname{\mathcal{D}}) \rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$.