  In this work, we study the notion of cofinal functor of $\infty$-bicategories
with respect to the theory of partially lax colimits. The main result of this
paper is a characterization of cofinal functors of $\infty$-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
$\infty$-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 $f$.
