# Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$

Proposition 6.3.3.13. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty$-categories. The following conditions are equivalent:

$(1)$

The functor $F$ is a reflective localization.

$(2)$

The functor $F$ admits a right adjoint and exhibits $\operatorname{\mathcal{D}}$ as the localization of $\operatorname{\mathcal{C}}$ with respect to some collection of morphisms $W$ of $\operatorname{\mathcal{C}}$.

$(3)$

The functor $F$ admits a fully faithful right adjoint $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$.

$(4)$

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 exhibits $G$ as a right adjoint of $F$.

$(5)$

The functor $F$ admits a right adjoint $G$ for which the composition $(F \circ G): \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{D}}$ is an equivalence of $\infty$-categories.

Proof. Note that any of these conditions guarantee that $F$ admits a right adjoint $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$. The equivalence $(1) \Leftrightarrow (3) \Leftrightarrow (4) \Leftrightarrow (5)$ follow by applying Corollary 6.2.2.13 to the functor $G$, and the implication $(1) \Rightarrow (2)$ is immediate. We will complete the proof by showing that $(2)$ implies $(4)$. Assume that $F$ exhibits $\operatorname{\mathcal{D}}$ as the localization of $\operatorname{\mathcal{C}}$ with respect to a collection of morphisms $W$, and let $\epsilon : F \circ G \rightarrow \operatorname{id}_{\operatorname{\mathcal{D}}}$ be the counit of an adjunction. We wish to show that $\epsilon$ is an isomorphism. 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

$\pi _0( \operatorname{Fun}(\operatorname{\mathcal{D}}, \operatorname{\mathcal{E}})^{\simeq } ) \xrightarrow {\circ [F]} \pi _0( \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{E}})^{\simeq } ),$

which follows immediately from our assumption on $F$. $\square$