David Yetter, in his query, seems to be making heavy weather of the composition of enriched functors. The composite of F:A --> B and G:B --> C is given on objects by GFa, while its "effect on maps" is the composite of the effects of F and of G as in A(a,a') ------> B(Fa,Fa') -----> C(GFa,GFa'). That is all. There is further a composition of profunctors (also called bimodules, or just modules), given by an evident co-end formula. Now every functor F determines a profunctor F*, where F*(a,b) is B(Fa,b); and the calculation David refers to (which is a simple application of the Yoneda isomorphism) is just the verification that (GF)* is canonically isomorphic to (G*)(F*). Of course this holds, in particular, for C-linear categories. New-Year greetings to all - Max Kelly.