2-Cartesian fibrations II: A Grothendieck construction for $\mathrm{\infty }$-bicategories

In this work, we conclude our study of fibred $\mathrm{\infty }$-bicategories by providing a Grothendieck construction in this setting. Given a scaled simplicial set $S$ (which need not be fibrant) we construct a 2-categorical version of Lurie’s straightening-unstraightening adjunction, thereby furnishing an equivalence between the $\mathrm{\infty }$-bicategory of 2-Cartesian fibrations over $S$ and the $\mathrm{\infty }$-bicategory of contravariant functors $S^{\mathrm{op}}\to \mathbb{B}\text{\hspace{0.17em}⁣}{\mathrm{icat}}_{\mathrm{\infty }}$ with values in the $\mathrm{\infty }$-bicategory of $\mathrm{\infty }$-bicategories. We provide a relative nerve construction in the case where the base is a 2-category, and use this to prove a comparison to existing bicategorical Grothendieck constructions.