On cofinal functors of $\mathrm{\infty }$-bicategories

In this work, we study the notion of cofinal functor of $\mathrm{\infty }$-bicategories with respect to the theory of partially lax colimits. The main result of this paper is a characterization of cofinal functors of $\mathrm{\infty }$-bicategories via generalizations of the conditions of Quillen’s Theorem A. As a key ingredient for the proof of our main theorem we produce for every functor of $\mathrm{\infty }$-bicategories $f:\mathbb{C}\to \mathbb{D}$ an outer 2-Cartesian fibration $\mathbb{F}(\mathbb{C})\to \mathbb{D}$ which we identify it as the free fibration on the functor $f$.