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

Corollary Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories and let $n$ be an integer. Then $F$ is categorically $n$-connective if and only if it factors as a composition

\[ \operatorname{\mathcal{C}}\xrightarrow {E} \operatorname{\mathcal{C}}' \xrightarrow {F'} \operatorname{\mathcal{D}}' \xrightarrow {G} \operatorname{\mathcal{D}} \]

where $E$ and $G$ are equivalences of $\infty $-categories and $F'$ is bijective on simplices of dimension $\leq n$. Moreover, we can arrange that $E$ and $F'$ are monomorphisms and that $G$ is a trivial Kan fibration.