Proposition 4.8.7.13. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories and let $n \geq 0$ be an integer. Suppose that $F$ is bijective on simplices of dimension $< n$ and surjective on simplices of dimension $n$. Then $F$ is categorically $n$-connective.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof. Note that $F$ is automatically essentially surjective (since it is surjective on objects). By virtue of Remark 4.8.7.5, it will suffice to show that for every pair of objects $X,Y \in \operatorname{\mathcal{C}}$, the map of Kan complexes
\[ F_{X,Y}: \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y ) \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{D}}}( F(X), F(Y) ) \]
is $(n-1)$-connective. This follows from Corollary 3.5.2.2, since $F_{X,Y}$ is bijective on simplices of dimension $< n-1$ and surjective on simplices of dimension $n$. $\square$