Given a functor \(H \colon \mathcal B\rightarrow\mathcal C\) of \(\infty\)-categories and a commuting square of \(\infty\)-categories 
Assume that for any \(b_0, b_1 \in \mathcal B\) in the image of \(\{0\} \times \mathcal A\rightarrow S^0 \times \mathcal A\rightarrow\mathcal B\) and \(\{1\} \times \mathcal A\rightarrow S^0 \times \mathcal A\rightarrow\mathcal B\), respectively, and any commuting square 
the space of dashed lifts is contractible. Then, the space of lifts of the square ([00GE]) is contractible.
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. ◻
Original source: arXiv:2401.02956v2
Original source · 2401.02956v2