In an \(\infty\)-category \(\mathcal V\) with a factorization system \((\mathcal L, \mathcal R)\), consider a commuting square 
and a further morphism \(Q\rightarrow A\) in \(\mathcal L\) so that also the composite \(Q\rightarrow C\) is in \(\mathcal L\). Let \(Q\rightarrow B|_{Q} \rightarrow B\) and \(Q \rightarrow D|_Q \rightarrow D\) denote the factorizations of the induced morphisms from \(Q\). Then, the map between spaces of (dashed) lifts 
is an equivalence. 39
Proof.
Let \(\mathcal V_{Q/^{\mathcal L}}\) denote the full subcategory of \(\mathcal V_{Q/}\) on the morphisms \(Q\rightarrow X\) which are in \(\mathcal L\). The factorization system induces a right adjoint of the inclusion \(\mathcal V_{Q/^{\mathcal L}} \hookrightarrow \mathcal V_{Q/}\) which sends \(Q\rightarrow X\) to its factorization \(Q\rightarrow X|_Q\). The statement then follows immediately from adjunction. ◻
Original source: arXiv:2401.02956v2
Original source · 2401.02956v2