Theorem 4.8.4.6. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories. Then $F$ is an equivalence of $\infty $-categories if and only if the induced map of Kan complexes $\theta : \operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}})^{\simeq } \rightarrow \operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{D}})^{\simeq }$ is a homotopy equivalence.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof. Assume that $\theta $ is a homotopy equivalence; we will show that $F$ is an equivalence of $\infty $-categories (the reverse implication is a special case of Proposition 4.5.1.22). Note that the map of cores $\operatorname{\mathcal{C}}^{\simeq } \rightarrow \operatorname{\mathcal{D}}^{\simeq }$ is a retract of $\theta $, and is therefore also a homotopy equivalence (Proposition 3.2.8.3). In particular, it is surjective on connected components, so that $F$ is essentially surjective. By virtue of Theorem 4.8.4.1, it will suffice to show that for every pair of objects $X,Y \in \operatorname{\mathcal{C}}$, the map $F_{X,Y}: \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{D}}}( F(X), F(Y) )$ is a homotopy equivalence. This follows by applying Proposition 3.3.7.1 to the commutative diagram of Kan complexes
\[ \xymatrix@C =50pt@R=50pt{ \operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}})^{\simeq } \ar [r] \ar [d] & \operatorname{Fun}( \Delta ^1,\operatorname{\mathcal{D}})^{\simeq } \ar [d] \\ \operatorname{Fun}( \operatorname{\partial \Delta }^1, \operatorname{\mathcal{C}})^{\simeq } \ar [r] & \operatorname{Fun}( \operatorname{\partial \Delta }^1, \operatorname{\mathcal{D}})^{\simeq } } \]
where the horizontal maps are homotopy equivalences (by assumption) and the vertical maps are Kan fibrations (Corollary 4.4.5.4). $\square$