The object representing a functor from a category C to Set is well understood, being unique if it exists up to unique isomorphism and all that. In the only case I'm familiar with, functors from k-Alg to Grp, the representing objects can be given a Hopf algebra structure with which to determine the group structure. Is there a general theory on how to give representing objects additional structure, when one is interested in functors to other categories than Set (of which the above example would be a particular case)? In the case I'm dealing with in particular, the terminal category has as objects affine spaces. If this is well covered in the literature, please forgive the question and just point me to a reference. Thanks very much! Allen K. ==============================================================================