Further to my previous posting, it occured to me that the second sentence of the following paragraph wasn't formulated correctly:
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 monomorphisms.
It should read: Having (1)-(3), ie an M-complete category C with M closed under composition, then the companion E of M is a class of epimorphisms iff C has equalizers and M contains all regular monomorphisms. Walter. 15-Oct-2002 15:26:37 -0300,2406;000000000001-00000000