  Presheaves on a small category are well-known to correspond via a category of
elements construction to ordinary discrete fibrations over that same small
category. Work of R. Par\'e proposes that presheaves on a small double category
are certain lax functors valued in the double category of sets with spans. This
paper isolates the discrete fibration concept corresponding to this presheaf
notion and shows that the category of elements construction introduced by
Par\'e leads to an equivalence of virtual double categories.
