None

Let C be any category, and A a set of arrows in C. Say that an object c of C _satisfies_ A if for every arrow p: a ---> a' in A, any arrow f: a ---> c factors through p as f = f'p. Denote by C:A the full subcategory on all the objects that satisfy "axioms" A. I believe that this was first done by Banaschewski and Herrlich Houston Math. J. (1977) 149-171. See also Section 22 of the book "Abstract & Concrete Categories" [Wiley Interscience]. George Strecker ++++++++++++++++++++++