Is the following known? (I don't know if I have actually proved it, but it looks good and I don't want to waste effort on the details if it is known, as seems fairly likely.) Suppose C is a category with a factorization system E/M and T is a triple on C, with category of algebras C^T, free and underlying functors F^T and U^T. Under reasonable hypotheses (complete and M-well-powered) there is a factorization system E^T/M^T on C^T in which M^T is U^{-1}(M) and E^T is the complementary class. Then there is a 1-1 correspondence between full subcategories of C^T closed under M^T-subobjects, products, and split epimorphisms; and triple maps T --> S for which F^T --> F^S lies in E^T. It is crucial, BTW, that the E's be epic and the M's monic.