Thomason's colimit theorem for the double category of elements

We show that, for any 2-category $C$ and 2-functor $F\text{\hspace{0.17em}⁣}:C\to Cat$, the double category of elements ${\iint }_{C}F$ introduced by Grandis and Paré satisfies a version of Thomason’s colimit theorem; that is, there is a weak homotopy equivalence $BhocolimF\simeq B({\iint }_{C}F)$.