Proposition 9.5.6.2 (06UW)

Proposition 9.5.6.2. Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be presentable $\infty $-categories. Then $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is a Bousfield localization functor if and only if it satisfies the following conditions:

$(1)$: The functor $F$ is cocontinuous: that is, it preserves small colimits.
$(2)$: The functor $F$ is a localization (in the sense of Definition 6.3.3.1): that is, it exhibits $\operatorname{\mathcal{D}}$ as a localization of $\operatorname{\mathcal{C}}$ with respect to some collection of morphisms.

Proof. By virtue of the adjoint functor theorem (Theorem 9.5.2.1), condition $(1)$ is equivalent to the requirement that $F$ admits a right adjoint $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$. In this case, $F$ is a Bousfield localization functor if and only if the functor $G$ is fully faithful, which is a reformulation of condition $(2)$ (see Proposition 6.3.3.7). $\square$