Recall from [GK17, Prop. 4.6] that any presentable \(\infty\)-category admits a factorization system of (\(n\)-connected, \(n\)-truncated) morphisms. We warn the reader that for any \(k \geq 1\) and any \(n > -2\), the (\(n\)-surjective, \(n\)-faithful) factorization system on \(\mathrm{Cat}_{(\infty, {k})}\) of Theorem 5.3.7 does not coincide with this (\(n\)-connected, \(n\)-truncated) factorization system induced from presentability of \(\mathrm{Cat}_{(\infty, {k})}\). This can already be seen in the case that \(k=1\): A functor \(\mathcal C\rightarrow\mathcal D\) of \((\infty,1)\)-categories is \(n\)-truncated if and only if it is so on spaces of objects and morphisms.29 Indeed, we have the diagram of irreversible implications 
for morphisms in \(\mathrm{Cat}_{(\infty, {1})}\). For example, a \((-1)\)-truncated functor of \((\infty,1)\)-categories, i.e. a monomorphism in \(\mathrm{Cat}_{(\infty, {1})}\), is a functor \(F\colon \mathcal C\rightarrow\mathcal D\) which for any two objects \(c, c' \in \mathcal C\) induces a \((-1)\)-truncated map \(\mathrm{Hom}_{\mathcal C}(c,c') \hookrightarrow \mathrm{Hom}_{\mathcal D}(Fc, Fc')\) which restricts to an equivalence between the full subspaces of isomorphisms \(\mathrm{Hom}_{\iota_0\mathcal C}(c,c') \rightarrow\mathrm{Hom}_{\iota_0\mathcal D}(Fc,Fc')\). In particular, any \((-1)\)-faithful, i.e. fully faithful, functor is \((-1)\)-truncated, and any \((-1)\)-truncated functor is \(0\)-faithful, but neither of these implications is reversible. In particular, a \(0\)-faithful functor \(F \colon \mathcal C\rightarrow\mathcal D\) does not necessarily exhibit \(\mathcal C\) as a subcategory of \(\mathcal D\) in the sense of subsection A.2.2.
Homwise iterating these observations, a similar diagram applies for \((\infty,k)\)-categories with \(k+1\) rows corresponding to the enrichment-depth at which functors between higher hom-categories are required to be truncated rather than faithful.