Proposition 6.3.3.2. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories and let $W$ be the collection of all morphisms $w$ of $\operatorname{\mathcal{C}}$ such that $F(w)$ is an isomorphism in $\operatorname{\mathcal{D}}$. Then $F$ is a localization functor if and only if it exhibits $\operatorname{\mathcal{D}}$ as a localization of $\operatorname{\mathcal{C}}$ with respect to $W$.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof. Assume that $F$ is a localization functor; we will show that it exhibits $\operatorname{\mathcal{D}}$ as a localization of $\operatorname{\mathcal{C}}$ with respect to $W$ (the reverse implication follows immediately from the definitions). Let $\operatorname{\mathcal{E}}$ be an $\infty $-category; we wish to show that composition with $F$ induces an equivalence of $\infty $-categories $\operatorname{Fun}( \operatorname{\mathcal{D}}, \operatorname{\mathcal{E}}) \rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}[W^{-1}], \operatorname{\mathcal{E}})$. Choose a collection of morphisms $W'$ of $\operatorname{\mathcal{C}}$ such that $F$ exhibits $\operatorname{\mathcal{D}}$ as a localization of $\operatorname{\mathcal{C}}$ with respect to $W'$. Note that $W'$ is contained in $W$, so that we can regard $\operatorname{Fun}( \operatorname{\mathcal{C}}[W^{-1}], \operatorname{\mathcal{D}})$ as a full subcategory of $\operatorname{Fun}( \operatorname{\mathcal{C}}[W'^{-1}], \operatorname{\mathcal{D}})$. It will therefore suffice to show that the composite functor
\[ \operatorname{Fun}( \operatorname{\mathcal{D}}, \operatorname{\mathcal{E}}) \xrightarrow { \circ F} \operatorname{Fun}( \operatorname{\mathcal{C}}[W^{-1}], \operatorname{\mathcal{E}}) \hookrightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}[W'^{-1}], \operatorname{\mathcal{E}}) \]
is an equivalence of $\infty $-categories, which follows from our assumption on the functor $F$. $\square$