Kerodon

Warning 6.3.3.12. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a localization functor of $\infty $-categories which is both reflective and coreflective. Then $F$ admits both a left adjoint $F^{L}: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ and a right adjoint $F^{R}: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$, which are automatically fully faithful (Proposition 6.3.3.6). Let $\operatorname{\mathcal{C}}^{L} \subseteq \operatorname{\mathcal{C}}$ be the full subcategory spanned by the essential image of the functor $F^{L}$, and define $\operatorname{\mathcal{C}}^{R} \subseteq \operatorname{\mathcal{C}}$ similarly. When viewed as abstract $\infty $-categories, $\operatorname{\mathcal{C}}^{L}$ and $\operatorname{\mathcal{C}}^{R}$ are equivalent (since they are both equivalent to the $\infty $-category $\operatorname{\mathcal{D}}$). Beware that they generally do not coincide as subcategories of $\operatorname{\mathcal{C}}$. See Warning 9.2.1.23.