Concerning the basic question raised by Hendryk Pfeiffer : If C is a symmetric monoidal category, it seems that there does not exist any category with finite limits whose representations in C are the C-enriched categories. An object in C is not suitable to serve as the Space of Objects of a C-category because the latter would require a well-behaved Diagonal Map. Unless of course C is cartesian. "Internal category" in a cartesian category is a coherent concept, as is "enriched category " in a general monoidal category because the latter is always presumed to be itself enriched in the cartesian category of sets where the spaces of objects are taken. I know of no way, for example, to define the enriched hom of a strong functor category without using the diagonal map on the space of objects of the domain because the very definition of natural transformation requires that the same thing ocurs in two different places. (Of course the problem is compounded for hypothetical enriched n-categories). The combination of internal and enriched features is implicit in mathematics. The most immediate way of making it explicit would seem to be simply to take as basic C->E , an enrichment of a monoidal category in a cartesian category. For any category based on that, any object I from E can parameterize its objects and any object V from C can parameterize its homs. (Giving a meaning to "smooth" parameterizations like these is after all the basic point of both enrichment and internalizations, because they do occur). Relative to such contexts, a suitable notion of theory can probably be found. There are plenty of examples: C= chain complexes in a topos E ... Are there publications which treat these issues ? A similar problem arises with the "comprehension scheme" when we try to set up an adjunction between modules (actions, presheaves) and "fibrations" . This does not seem routine when not only the fibers but also the base and the hypothetical total are to be, for example , Ab-enriched. Is there a coherent approach to this ? Note that for example the "trivial extension" idea of adjointing infinitesimals serves well as a way of making modules into rings (additive categories) but does not appear as any kind of cousin to the notion of fibered category.