Discrete Double Fibrations

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.