On orthogonal factorization systems and double categories

We prove that the $\mathrm{\infty }$-category of orthogonal factorization systems embeds fully faithfully into the $\mathrm{\infty }$-category of double $\mathrm{\infty }$-categories. Moreover, we prove an (un)straightening equivalence for double $\mathrm{\infty }$-categories, which restricts to an (un)straightening equivalence for op-Gray fibrations and curved orthofibrations of orthogonal factorization systems.