In our (John Kennison, Bob Raphael, and I) work, the following condition has arisen. Has anyone seen or named it? Say that an object E of a category is a ???? if it is a cogenerator and if whenever f: A ---> B is not an epimorphism and g: B ---> C is a regular monomorphism, then there are two maps h,k: C ---> E s.t. hg is unequal to kg, while hgf = kgf. This is related to the questions Paul Taylor and I have raised recently. Theorem. If E satisfies ????, then whenever A has an extremal monomorphism into a power of E, then A ---> TA ===> T^2A is an equalizer where T is the triple from the adjoint pair Hom(-,E) and E^{(-)}. What's interesting is that while it is obvious that any injective cogenerator satisfies ????, it is also the case that any cogenerator that contains an injective cogenerator also satisfies ????. Thus, in completely regular spaces, the interval is a cogenerator and both it and the real line (and many, many other spaces) also satisfy ????. Michael