Proposition 6.3.1.20. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a morphism of simplicial sets, where $\operatorname{\mathcal{D}}$ is an $\infty $-category, and let $W$ be the collection of all edges of $\operatorname{\mathcal{C}}$. The following conditions are equivalent:
The morphism $F$ exhibits $\operatorname{\mathcal{D}}$ as a localization of $\operatorname{\mathcal{C}}$ with respect to $W$.
The $\infty $-category $\operatorname{\mathcal{D}}$ is a Kan complex and $F$ is a weak homotopy equivalence of simplicial sets.
Proof.
We first prove that $(2)$ implies $(1)$. Assume that $\operatorname{\mathcal{D}}$ is a Kan complex and that $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is a weak homotopy equivalence; we wish to show that $F$ exhibits $\operatorname{\mathcal{D}}$ as a localization of $\operatorname{\mathcal{C}}$ with respect to $W$. By virtue of Proposition 6.3.1.13, it will suffice to show that for every $\infty $-category $\operatorname{\mathcal{E}}$, composition with $F$ induces a homotopy equivalence of Kan complexes $\theta : \operatorname{Fun}(\operatorname{\mathcal{D}}, \operatorname{\mathcal{E}})^{\simeq } \rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}[W^{-1}], \operatorname{\mathcal{E}})^{\simeq }$. Since $\operatorname{\mathcal{D}}$ is a Kan complex, Proposition 4.4.3.22 allows us to identify $\theta $ with the canonical map
\[ \operatorname{Fun}( \operatorname{\mathcal{D}}, \operatorname{\mathcal{E}}^{\simeq }) \xrightarrow { \circ F} \operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}}^{\simeq } ), \]
which is a homotopy equivalence by virtue of our assumption that $F$ is a weak homotopy equivalence.
We now show that $(1)$ implies $(2)$. Assume that $F$ exhibits $\operatorname{\mathcal{D}}$ as a localization of $\operatorname{\mathcal{C}}$ with respect to $W$. Invoking Remark 6.3.1.16, we deduce that $F$ is a weak homotopy equivalence. We wish to show that $\operatorname{\mathcal{D}}$ is a Kan complex. Choose a weak homotopy equivalence $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$, where $\operatorname{\mathcal{E}}$ is a Kan complex (Corollary 3.1.7.2). Then the composite map $(G \circ F): \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{E}}$ is also a weak homotopy equivalence (Remark 3.1.6.16). Invoking the implication $(2) \Rightarrow (1)$, we conclude that $G \circ F$ exhibits $\operatorname{\mathcal{E}}$ as a localization of $\operatorname{\mathcal{C}}$ with respect to $W$. It follows from Remark 6.3.1.19 that $G$ is an equivalence of $\infty $-categories. Since $\operatorname{\mathcal{E}}$ is a Kan complex, it follows that the $\infty $-category $\operatorname{\mathcal{D}}$ is also a Kan complex (Remark 4.5.1.21).
$\square$