Given a scaled simplicial set $S$ we construct a 2-categorical version of the straightening-unstraightening adjunction furnishing an equivalence between the $\infty$-bicategory of outer 2-Cartesian fibrations over $S$ and the $\infty$-bicategory of contravariant functors $S^{\operatorname{op}}\to \mathbb{B}\!\operatorname{icat}_{\infty}$ with values in the $\infty$-bicategory of $\infty$-bicategories. We use this technology to characterize cofinal functors of $\infty$-bicategories via variants of the conditions of Quillen's Theorem A.