The composition of F and G is then given by
\integral^c Hom(F(-),c) \otimes Hom(G(c),-) \cong Hom(G(F(-)),-).
This follows from the enriched Yoneda Lemma. The setting is as follows. It is tensor product of (bi)modules which is given by the coend formula. Modules (= profunctors = distributors) between V-categories are the arrows for a bicategory Mod(V) with this tensor product as composition. Each V-functor F becomes a module in two ways: F_* and F^*. The latter involves the Hom(F(a),c) as required by David. Each process gives a locally fully faithful embedding of V-Cat in V-Mod (Yoneda's Lemma) with appropriate variances. Moreover, F^* is right adjoint to F_* in the bicategory Mod(V). A good reference is Lawvere's paper "Metric spaces, generalised logic, and closed categories" Rend. Sem. Mat. Fis. Milano 43 (1974) 135-166. Best wishes for the New Year, Ross