Name for this concept?

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. Bill Rowan ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++