Some not very exciting developments: If F: C -> D is a functor, then the question was about when F*: S^D -> S^C was full and faithful. (Incidentally, it took a lot of headache before I convinced myself that "fully faithful" just meant "full and faithful". I couldn't find the phrase defined in any of the standard texts. Is its mellifluousness really enough to justify its use?) My conjecture was that that this happens iff F is an epimorphism of categories. This can't be correct. An epimorphism must be surjective on objects (otherwise you can map the objects not in the image to distinct isomorphic copies), but if F is any equivalence then F* is as well, and hence full and faithful. (So my account of Mitchell's results for ringoids as opposed to rings was probably oversimplified.) I assume that "epimorphism", with its demands of on-the-nose equality, is simply not a good notion in the 2-categorical context of categories. In the monoid case, we do have an implication in at least one direction (adapted from the argument for rings): Suppose C and D are monoids. If F* is full and faithful, then F is an epimorphism (of monoids). Proof: Let G,H: D => E with F;G = F;H. E is a D-set, with action ed = e.H(d) Consider the function G: D -> E. This is a homomorphism of C-sets, for G(dc) = G(d).G(F(c)) = G(d).H(F(c)) = G(d)F(c) = G(d)c Hence by fullness of F* (note that faithfulness is automatic in this context), it is a homomorphism of D-sets, so for any d in D we have G(d) = G(1d) = G(1)d = 1.H(d) = H(d) Remaining questions: * Is the converse true in the monoid case? * Is there a 2-categorical generalization of epi that (includes equivalences and) restores the original conjecture? Steve Vickers. ==============================================================================