In this paper, Frege's notion of "unsaturated", and more precisely the notion of "concept", is analysed from the point of view of its connections with the parallel notion of "object". The variety of these connections makes it evident that they cannot be all reduced to the fundamental relation between objects and concepts - "falling under" - which, presumably, the notion of "unsaturated" has to be based on and ended with. Then the issue of a general characterization of unsaturated is discussed and various possible ways are considered that turn out to be closed. Finally, we propose a characterization by putting forward a theory that implicitly and in general fixes of relations that can legitimately exist between objects and concepts. We call such kind of relations "copular" relations. Arguments are given that show that this notion of copularity is logically founded and not merely conventional.