ScalingStacks

Since the inclusion \(\mathrm{Cat}_{(\infty, {k})} \hookrightarrow \mathrm{Cat}_{(\infty, {n})}\) preserves pullbacks, and preserves \(n\)-factorizations and hence commutes with \(\tau_n\), it suffices to prove the statement for \(n=k\). We induct on \(n\geq 0\). The base case \(n=0\) is immediate. For general \(n\), we prove that the underlying diagram of spaces is a pullback square and that for each \(c, c' \in \mathcal C\) the induced square of \((\infty,n-1)\)-categories is a pullback square. Using lemma 5.4.7, the latter square is a pullback square by induction.