Dear Bill, you have asked whether somebody has clarified the notion of enriched internal category. I don't know of any written account but it most certainly has been studied by Jean B\'enabou in his work on fibred categories. I think one should start as follows. Consider some base category BB with finite limits and consider some fixed fibred monoidal category C : CC -> BB. (The latter notion (due to B\'enabou) I think can be found in some seminar reports by Gouzou & Grunig from the 70ies. The idea is to consider a cartesian functor \otimes : C x_BB C --> C together with cartesian natural transformations \alpha, \lambda, \rho satisfying literally the same coherence conditions usually required for ordinary monoidal cats. Similarly, one gets a notion of fibred symmetric monoidal category etc.) Now given a fibration P : XX --> BB one may ask what it means that P is enriched over C ? An essential ingredient of such an enrichment should consist of a cartesian functor H : P^op x P ---> C whose I-th component enriches P(I) in C(I). The tricky thing is to conceptualize composition in the fibred enriched category. I think it must be something like a cartesian natural transformation c_{X,Y,Z} : H(X,Y) \otimes H(Y,Z) ----> H(X,Z) satisfying associativity requirements etc. But the problem is that Y has two occourrences, one co- and one contravariant. Thus, H(X,Y) \otimes H(Y,Z) should rather be replaced by a coend \int_Y H(X,Y) \otimes H(Y,Z) (\int stands for the integral symbol) Thus, one has to assume that P is locally small and that C is a cocomplete fibration. I am afraid one has to get into such a mess as one cannot simply consider a H : |P|^op x |P| ---> C with |P| the "presheaf of objects" because of the "equality of objects" problem. Hm, I see when starting to write things down I run into problems. At home I will have a look at this rapport by Gouzou & Grunig and hope that I can tell you more. Best, Thomas