In reply to Stefan's message, it is certainly possible to define V-categories for a general promonoidal V. Let V be a promonoidal category, with tensor functor P: C^op x C^op x C -> Set, and unit functor J: C -> Set. A V-category C consists of: a set obC of objects; a hom object C(A,B) in V for every (A,B) in obC^2; an identity element id_A in J(C(A,A)) for every A in obC; a composition element M_A,B,C in P(C(B,C),C(A,B),C(A,C)) for every (A,B,C) in obC^3; subject to associativity and unit axioms that can be expressed as the commutativity of three smallish diagrams that involve coends. I think it's equivalent to ask for a ([V,Set]^op)-category all of whose hom objects are representable. (Of course it is possible that this is not what Kelly et al. had in mind.) Robin 30-Mar-2005 11:10:59 -0400,2336;000000000001-00000000