Fix any \(j \geq k \geq 0\) and \(n \geq -2\). By the description of the generators in theorem 5.3.7.([00A7]), and their truncations in observation 5.1.7, we see that the inclusion \(\mathrm{Cat}_{(\infty, {k})} \xhookrightarrow{i_j} \mathrm{Cat}_{(\infty, {j})}\) preserves and detects the (\(n\)-surjective, \(n\)-faithful) factorization system.30 In particular, a map of spaces \(f \colon X \rightarrow Y\) is \(n\)-connected (or \(n\)-truncated) if and only if it is \(n\)-surjective (or \(n\)-faithful) as a map of \((\infty, k)\)-categories for any \(k \geq 0\).
It follows that \(n\)-factorizations in \(\mathrm{Cat}_{(\infty, {k})}\) remain so in \(\mathrm{Cat}_{(\infty, {j})}\). It also follows that the left adjoint \(\mathrm{Cat}_{(\infty, {j})} \xrightarrow{|-|_k} \mathrm{Cat}_{(\infty, {k})}\) preserves the notion of \(n\)-surjectivity and that the right adjoint \(\mathrm{Cat}_{(\infty, {j})} \xrightarrow{\iota_k} \mathrm{Cat}_{(\infty, {k})}\) preserves the notion of \(n\)-faithfulness.