Ross Street asks Michael Barr if he knows "of a diagram lemma which could be proved significantly quicker using the embedding theorem than by the generalized-element/relations technology". Well, I don't exactly, but one of the earliest uses of Barr's embeddings of regular categories was in my American Journal of Mathematics paper on Simplicial Homotopy in an Exact Category (1977, vol 99, pp 1193-1204). I wanted to prove the homotopy exact sequence of a Kan fibration of sheaves of simplicial sets and Barr's results were exactly what I needed. All this has since been improved by using closed model structures and a less naive definition of fibration, but the "embedding/technology" Ross mentions was all we had available--in sufficient generality to cover my situation--at the time. Don Van Osdol