Regular and exact (virtual) double categories

We propose definitions of regular and exact (virtual) double categories, proving a number of results which parallel many basic results in the theory of regular and exact categories. We show that any regular virtual double category admits a factorization system which generalizes the factorization of a functor between categories into a bijective-on-objects functor followed by a fully-faithful functor. Finally, we show that our definition of exact double category is equivalent to an axiom proposed by Wood, and very closely related to the "tight Kleisli objects" studied by Garner and Shulman.