Consider a morphism \(\mathcal C\xrightarrow{F} \mathcal D\) in \(\mathrm{Cat}_{(\infty, {k})}\) for some \(k \geq 0\) and let \(n\geq -2\).
We declare that any \(F\) is \((-2)\)-surjective and that \(F\) is \((-2)\)-faithful if it is an equivalence.
If \(k=0\), we say that \(F\) is \(n\)-surjective if it is \(n\)-connected and \(n\)-faithful if it is \(n\)-truncated.27
For \(n > -2\) and \(k > 0\), we inductively define \(F\) to be
\(n\)-surjective if it is surjective on objects and for every \(c,c' \in \mathcal C\) the morphism \(\underline{\mathrm{Hom}}_\mathcal C(c,c') \rightarrow\underline{\mathrm{Hom}}_\mathcal D(Fc,Fc')\) in \(\mathrm{Cat}_{(\infty, {k-1})}\) is \((n-1)\)-surjective, and
\(n\)-faithful if for every \(c,c' \in \mathcal C\) the morphism \(\underline{\mathrm{Hom}}_\mathcal C(c,c') \rightarrow\underline{\mathrm{Hom}}_\mathcal D(Fc,Fc')\) in \(\mathrm{Cat}_{(\infty, {k-1})}\) is \((n-1)\)-faithful.