Kerodon

Variant 6.3.3.11. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories. We say that $F$ is a coreflective localization functor if it admits a fully faithful left adjoint $\operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$. Equivalently, $F$ is a coreflective localization functor if the opposite functor $F^{\operatorname{op}}: \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{D}}^{\operatorname{op}}$ is a reflective localization functor (see Remark 6.2.1.7). It follows from Proposition 6.3.3.6 that every coreflective localization functor is a localization functor (in the sense of Definition 6.3.3.1).