Well, it may be that completeness is not required, but I am not now interested in hypothesis chopping. Maybe later. As for whether it is necessary that E consist of epis, if it doesn't, then not every regular monic is in M and then the category of S-algebras is not even closed under equalizers, therefore not reflective. While this case, might have some interest, I am trying to get a clean statement with a direct proof. I have come to the conclusion that a better theorem might just use an E/M factorization in the category of algebras. For example, by taking epi/regular mono in the category of monoids, you get that groups are an example. And indeed groups are closed in monoids under regular subobjects. You don't get every full subcategory, since you have rational vector spaces are a full subcategory of abelian groups, but it is possible to show that if F^T --> F^S is epic, then the you always get a full subcategory. Michael 13-Oct-2002 19:55:44 -0300,1490;000000000000-00000000