Question on subobject classifiers

I am reviewing a paper whose main result sounds like it ought to be known. Is it and can anyone give me a citation? Let E be a cocomplete category with a small dense subcategory C (so every e of E is the colimit of C/e --> E). Then an object \Omega of E is a subobject classifier in E iff it represents the subobject functor restricted to C. ------------------------------------------------------------------- If a society puts up with bad plumbers because plumbing is such a low calling, and if it puts up with bad philosophers because philosophy is such a high calling---then neither its pipes nor its theories will hold water. --- Slight paraphrase of former HEW secretary John Gardner