Comparison of Waldhausen constructions

In previous work, we develop a generalized Waldhausen S∙-construction whose input is an augmented stable double Segal space and whose output is a unital 2-Segal space. Here, we prove that this construction recovers the previously known S∙-constructions for exact categories and for stable and exact (∞,1)-categories, as well as the relative S∙-construction for exact functors.