Kerodon

Remark 9.5.6.6. Let $\operatorname{\mathcal{D}}$ be a presentable $\infty $-category. We can always find Bousfield localization functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$, where $\operatorname{\mathcal{C}}= \operatorname{Fun}( \operatorname{\mathcal{C}}_0^{\operatorname{op}}, \operatorname{\mathcal{S}})$ arises as the cocompletion of an essentially small $\infty $-category $\operatorname{\mathcal{C}}_0$. By virtue of Example 9.5.6.4, this is equivalent to the problem of finding a dense functor $f: \operatorname{\mathcal{C}}_0 \rightarrow \operatorname{\mathcal{D}}$, where $\operatorname{\mathcal{C}}_0$ is essentially small. If $\operatorname{\mathcal{D}}$ is $\kappa $-presentable, we can take $\operatorname{\mathcal{C}}_0 = \operatorname{\mathcal{D}}_{< \kappa }$ to be the full subcategory of $\operatorname{\mathcal{D}}$ spanned by the $\kappa $-compact objects (and $f$ to be the inclusion functor).