Proof.
Since \({\sf pt}\) and \([1]\) generate \(\mathrm{Cat}_{\infty}\) under colimits, it suffices to show that for every \(a\in \mathcal A\) and every arrow \([1] \xrightarrow{\{f\}} \mathcal A\), the induced total squares 
have contractible spaces of lifts. Contractibility of the spaces of lifts of the former square follows immediately from assumption, and for the latter square is a straight-forward computation assuming ([00GC]) is an equivalence. ◻