From (2) you necessarily have that M contains the isos and is contained in
Just to echo some of Michael's points, here is how I would look at his problem. Call (as I have done since the seventies) a category C, for any class M of morphisms, M-complete if (1) a pullback of an M along any morphism exists and is in M, (2) an intersection (=multiple pullback) of any-size family of M's with common codomain exists and (its diagonal) is in M. (Of course, "small" suffices when you assume C to be M-wellpowered.) the class of monos; the latter fact comes from the observation that when you take a multiple pullback of as many copies of a given morphism as the size of the category, that morphism got to be monic: see my paper with Reinhard Boerger in Math. Z. 160 (1978) 135-138. Now, add to (1) and (2) the condition that (3) M is closed under composition, then M is the right factor of a factorization system in C, and except for the existence assumptions in (1), (2), all conditions are necessary. Actually, you not only have orthogonal factorizations of morphisms, but of any-size families of morphisms with common codomain, and that statement is fully equivalent to (1)-(3). From this angle you see that you also get the existence of equalizers from (1) and (2), which is of interest when you want to prove that the left companion E of M is a class of epimorphisms iff M contains all regular monos. In Michael's situation where you have a monadic (or tripleable, if you insist) category A over C, with its limit-creating forgetful functor U, if C satisfies (1)-(3) for M, A will satisfy the same conditions for U^{-1}M, and you are in business. All this is of course old stuff of more than 20 years ago (see, for example JPAA 15 (1979) 53-73), but the correspondence Michael indicates I haven't seen before in this form at all. However, I am wondering how his correspondence relates to the almost equally old Cassidy-Hebert-Kelly paper and its predecessors (Gabriel, Ringel)? Regards, Walter.