Dear Bill, I have now looked up the paper I mentioned in my previous mail. It is a report form the Seminaire B'enabou authored by Gouzou and Grunig entiteled Fibrations Relatives dated from Novemeber 1976. In the first chapter they introduce fibred monoidal categories U : UU -> BB. In the second chapter they define UU-enriched BB-categories as follows: an object C_0 in BB, a family C_1 in U(C_0 x C_0), a map \eta : 1_{C_0} --> \Delta^*C_1 (here \Delta is the diagonal on C_0) providing identities and a map \mu : p_3^*H \otimes p_1^*C_1 --> p_2^*H in U(C_0xC_0xC_0) providing multiplication. These data have to satisfy the obvious identities. The problem I ran into in my previous mail is avoided here by assuming that C_0 is a small discrete fibration in Psh(BB) and thus has definable equality. It is not clear to me how to extend their approach to enriched fibred categories with a non-small set of objects. Best, Thomas PS The report of Gouzou and Grunig contains much more material (about 90 pages) I haven't looked at yet.