Regarding Paul Taylor's question, here are some useful references that contain answers: G.M. Kelly, Monomorphisms, epimorphisms, and pullbacks, J. Austral. Math Soc. 9 (1969) 124-142. (Mentions an example of a complete,cocomplete, wellpowered, cowellpowered additive category due to Isbell in which reg. epis do not compose. The following articles deal with long chains of regular epis; this is a selection only:) J. MacDonald and A. Stone,The tower and regular decomposition , Cahiers TGDC 23 (1983) 197-213 R. Bo"rger and W. Tholen,Total categories and solid functors, Can. J. Math 42(1990) 213-229 (see sections 4 and5) J. Adamek and W. Tholen, Total categories with generators, J. of Alg. 133 (1990) 63-78 (see section 1) R. Bo"rger, Making factorizations compositive, Comm. Math. Univ. Carolin. (1991) 749-759 (see section 3). The last article mentions one remaining problem on the composition- cancellation behaviour of regular epis in a category (with kernel pairs and their coequalizers, say): consider the properties (A) and (B): (A) if e is split epi and d is regular epi, then the composite ed of first d and then e is regular epi, (B) if a composite ed is regular epi, then e is regular epi one always has that if (A) holds in our category, then also (B) holds. But what about the converse? (Note that neither property holds true in general, already Kelly's paper contains a counter-example.) Regards, Walter Tholen +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++