Comparison of Waldhausen constructions

In previous work, we developed a generalizedWaldhausen S-center dot-construction whose input is an augmented stable double Segal space and whose output is a 2-Segal space. Here, we prove that this construction recovers the previously known S-center dot-constructions for exact categories and for stable and exact (infinity,1)-categories, as well as the relative S-center dot-construction for exact functors.