Remark 6.3.3.4. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories. Using Corollary 6.2.2.19 to the right adjoint of $F$, we see that the following conditions are equivalent:
The functor $F$ is a reflective localization: that is, it admits a fully faithful right adjoint $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$.
There exists a functor $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ and a natural isomorphism $\epsilon : F \circ G \xrightarrow {\sim } \operatorname{id}_{\operatorname{\mathcal{D}}}$ which is the counit of an adjunction between $F$ and $G$.
The functor $F$ admits a right adjoint $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ for which the composition $(F \circ G): \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{D}}$ is an equivalence of $\infty $-categories.
Moreover, if these conditions are satisfied, then the essential image of $G$ is a reflective subcategory of $\operatorname{\mathcal{C}}$.