Definition 8.4.0.1. Let $h: \operatorname{\mathcal{C}}\rightarrow \widehat{\operatorname{\mathcal{C}}}$ be a functor of $\infty $-categories. We will say that $h$ exhibits $\widehat{\operatorname{\mathcal{C}}}$ as a cocompletion of $\operatorname{\mathcal{C}}$ if the following conditions are satisfied:
- $(1)$
The $\infty $-category $\widehat{\operatorname{\mathcal{C}}}$ admits small colimits.
- $(2)$
Let $\operatorname{\mathcal{D}}$ be an $\infty $-category which admits small colimits and let $\operatorname{Fun}'( \widehat{\operatorname{\mathcal{C}}}, \operatorname{\mathcal{D}})$ denote the full subcategory of $\operatorname{Fun}( \widehat{\operatorname{\mathcal{C}}}, \operatorname{\mathcal{D}})$ spanned by those functors which preserve small colimits. Then precomposition with $h$ induces an equivalence of $\infty $-categories $\operatorname{Fun}'( \widehat{\operatorname{\mathcal{C}}}, \operatorname{\mathcal{D}}) \rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$.