Proof.
We show that the functor \[ \textrm{Ar}^{(n-1)}(\mathrm{Cat}_{(\infty, {k})}) \rightarrow\mathrm{Cat}_{(\infty, {k})} \times_{\mathrm{Cat}_{({n}, {k})}} \textrm{Ar}^{(n-1)}(\mathrm{Cat}_{({n}, {k})})\] is surjective and fully faithful.
Surjectivity amounts to the following: For any \((\infty,k)\)-category \(\mathcal D\) equipped with a \((n-1)\)-faithful functor \(\mathcal C' \rightarrow\tau_n \mathcal D\) from an \((n,k)\)-category \(\mathcal C'\), there exists an \((\infty,k)\)-category \(\mathcal C\) and a \((n-1)\)-faithful functor \(\mathcal C\rightarrow\mathcal D\) which is sent to \(\mathcal C' \rightarrow\tau_n \mathcal D\) under \(\tau_n\).
Define \(\mathcal C\) to be the pullback in \(\mathrm{Cat}_{(\infty, {k})}\) 
Since the right class of a factorization system is preserved under pullback, and since \(\mathcal C' \rightarrow\tau_n \mathcal D\) is \((n-1)\)-faithful, so is its pullback \(\mathcal C\rightarrow\mathcal D\). We will now prove by induction on \(n\geq 0\) that the functor \(\tau_n \mathcal C\rightarrow\mathcal C'\) adjunct to \(\mathcal C\rightarrow\mathcal C'\) is an equivalence, proving surjectivity of ([00BV]). The base case \(n=0\) is immediate. In general, we will show that \(\tau_n \mathcal C\rightarrow\mathcal C'\) is surjective on objects and fully faithful. Since \(\mathcal D\rightarrow\tau_{n}\mathcal D\) is surjective on object (in fact \((n-1)\)-surjective), the pullback \(\mathcal C\rightarrow\mathcal C'\) is surjective on objects. Since \(\mathcal C\rightarrow\mathcal C'\) factors as \(\mathcal C\rightarrow\tau_n \mathcal C\rightarrow\mathcal C'\), it follows that also \(\tau_n \mathcal C\rightarrow\mathcal C'\) is surjective on objects. Fully faithfulness of \(\tau_n \mathcal C\rightarrow\mathcal C'\) follows by induction using lemma 5.4.7.
We now prove that ([00BV]) induces an equivalence on the hom-space between any pair of objects \(\{\mathcal C_1 \rightarrow\mathcal D_1\}, \{\mathcal C_2 \rightarrow\mathcal D_2\} \in\ \textrm{Ar}^{(n-1)}(\mathrm{Cat}_{(\infty, {k})})\), and hence that ([00BV]) is fully faithful. Unwinding the hom-spaces in the relevant arrow categories, this is equivalent to the statement that for any fixed functor \(G\colon \mathcal D_1 \rightarrow\mathcal D_2\) of \((\infty, k)\)-categories, the map of spaces of dashed lifts 
is an equivalence. This follows immediately from proposition 5.6.1. ◻