  In this paper we introduce a description of ordered groupoids as a particular
type of double categories. This enables us to turn Lawson's correspondence
between ordered groupoids and left-cancellative categories into a
biequivalence. We use this to identify which ordered functors are maps of sites
in the sense that they give rise to geometric morphisms between the induced
sheaf categories, and establish a Comparison Lemma for maps between Ehresmann
sites.
