Remark 9.5.7.9 (Presentations of Cocomplete $\infty $-Categories). In the situation of Corollary 9.5.7.8, we have a Bousfield localization functor $F: \widehat{\operatorname{\mathcal{K}}} \rightarrow \operatorname{\mathcal{C}}$, which restricts to a dense functor $f: \operatorname{\mathcal{K}}\rightarrow \operatorname{\mathcal{C}}$ (Example 9.5.6.4). For every presentable $\infty $-category $\operatorname{\mathcal{D}}$, Theorem 8.4.0.3 and Proposition 9.5.7.2 imply that the functor
is fully faithful, and that its essential image consists of those functors $T: \operatorname{\mathcal{K}}\rightarrow \operatorname{\mathcal{D}}$ for which the cocontinuous extension $\widehat{T}: \widehat{\operatorname{\mathcal{K}}} \rightarrow \operatorname{\mathcal{D}}$ carries each element of $W$ to an isomorphism in $\operatorname{\mathcal{D}}$. We will see later that this holds more generally when the $\infty $-category $\operatorname{\mathcal{D}}$ is cocomplete (Corollary 9.6.9.26). Stated more informally, an $\infty $-category $\operatorname{\mathcal{C}}$ is presentable if and only if it can be obtained from a small $\infty $-category $\operatorname{\mathcal{K}}$ by freely adjoining small colimits (to obtain the $\infty $-category $\widehat{\operatorname{\mathcal{K}}}$) and then forcing a small collection of morphisms $W$ to become isomorphisms (while remaining in the setting of cocomplete $\infty $-categories).