Re: Name for this concept?

Bill Rowan asks: "What do you call it if you have a category C, and you have a class X of arrows of C such that if x in X, then gxf in X for all composable isomorphisms f and g. A functor C->C' which is one leg of an equivalence takes such a set to another one X' in C', and any functor which is the other leg of the equivalence takes X' back to X." Well, I have called it an IDEAL; see G.M.Kelly, On the radical of a category, Jour. Austral. Math. Soc. 4 (1964), 299-307 and G.M.Kelly and F.W.Lawvere, On the complete lattice of essential localizations, Bull. Soc. Math. Belgique 41 (1989), 289-319. Max Kelly. ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++