Has anyone seen this condition?

Has anyone seen this condition?

[Michael Barr]

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