In previous work, we develop a generalized Waldhausen $S_{bullet}$-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 $S_{bullet}$-constructions for exact categories and for stable and exact $(infty,1)$-categories, as well as the relative $S_{bullet}$-construction for exact functors.
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