  It is known by results of Dyckerhoff-Kapranov and of
G\'alvez--Carrillo-Kock-Tonks that the output of the Waldhausen S.-construction
has a unital 2-Segal structure. Here, we prove that a certain S.-functor
defines an equivalence between the category of augmented stable double
categories and the category of unital 2-Segal sets. The inverse equivalence is
described explicitly by a path construction. We illustrate the equivalence for
the known examples of partial monoids, cobordism categories with genus
constraints and graph coalgebras.
