Proposition 6.3.3.6. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories. The following conditions are equivalent:
The functor $F$ is a reflective localization (in the sense of Definition 6.3.3.3): that is, it admits a fully faithful right adjoint.
The functor $F$ is a localization functor (in the sense of Definition 6.3.3.1) which admits a right adjoint $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$.
Proof. We first show that $(2)$ implies $(1)$. Let $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ be a right adjoint to $F$ and let $\epsilon : F \circ G \rightarrow \operatorname{id}_{\operatorname{\mathcal{D}}}$ be the counit of an adjunction. We wish to show that, if $F$ is a localization functor, then $\epsilon $ is an isomorphism (Remark 6.3.3.4). By virtue of Proposition 6.1.4.7 (applied to the opposite of the $2$-category $\mathrm{h}_{2} \mathit{\operatorname{\mathbf{QCat}}}$), it will suffice to show that for any $\infty $-category $\operatorname{\mathcal{E}}$, precomposition with the isomorphism class $[F] \in \pi _0( \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})^{\simeq } )$ induces a monomorphism
which follows immediately from our assumption on $F$.
We now prove the converse. Assume that $F$ is a reflective localization functor and let $W$ be the collection of all morphisms $w$ of $\operatorname{\mathcal{C}}$ such that $F(w)$ is an isomorphism in $\operatorname{\mathcal{D}}$; we will show that $F$ exhibits $\operatorname{\mathcal{D}}$ as a localization of $\operatorname{\mathcal{C}}$ with respect to $W$. Fix an $\infty $-category $\operatorname{\mathcal{E}}$, so that precomposition with $F$ induces a function
Let $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ and $\epsilon : F \circ G \rightarrow \operatorname{id}_{\operatorname{\mathcal{D}}}$ be as above, so that precomposition with $G$ induces a function
We will show that $\psi $ is inverse to $\varphi $. By assumption, the natural transformation $\epsilon $ is an isomorphism. It follows that, for every functor $E: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$, $\epsilon $ induces an isomorphism $E \circ F \circ G \xrightarrow {\sim } E$. Passing to isomorphism classes, we obtain an equality $[E] = [E \circ F \circ G] = \psi ( [ E \circ F ] ) = (\psi \circ \varphi )([E])$. Allowing $E$ to vary, we conclude that $\psi \circ \varphi $ is the identity on $\pi _0( \operatorname{Fun}( \operatorname{\mathcal{D}}, \operatorname{\mathcal{E}})^{\simeq } )$.
We now complete the proof by showing that $\varphi \circ \psi $ is the identity on $\pi _0( \operatorname{Fun}( \operatorname{\mathcal{C}}[W^{-1}], \operatorname{\mathcal{E}})^{\simeq } )$. Let $H: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{E}}$ be a functor of $\infty $-categories which carries each morphism of $W$ to an isomorphism in $\operatorname{\mathcal{E}}$; we wish to show that $H$ is isomorphic to $H \circ G \circ F$. Choose a unit map $\eta : \operatorname{id}_{\operatorname{\mathcal{C}}} \rightarrow G \circ F$ which is compatible with $\epsilon $ up to homotopy. For every object $C \in \operatorname{\mathcal{C}}$, Remark 6.3.3.5 guarantees that the morphism $\eta _{C}: C \rightarrow (G \circ F)(C)$ belongs to $W$, so that $H(\eta _{C} )$ is an isomorphism in the $\infty $-category $\operatorname{\mathcal{E}}$. Allowing the object $C$ to vary (and invoking the criterion of Theorem 4.4.4.4), we conclude that $\eta $ induces an isomorphism from $H$ to $H \circ G \circ F$ in the functor $\infty $-category $\operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}})$. $\square$