An elementary correspondence between double exact sequences and Bass--Swan generators

A double exact sequence is one of the generator types used in the paper's description of the relevant $K$-theory group, while a Bass--Swan generator is the corresponding generator in the alternative presentation. Correspondence conjecture. There should be a recipe to map a double exact sequence to a Bass--Swan generator. It should be possible to check in an elementary (but probably complicated) fashion that composing both maps either way yields the identity map. Establishing such mutually inverse maps would provide an elementary equivalence between the two generator-and-relation descriptions.