Given a topos and an object in , one notices that the slice category is a symmetric monoidal category; hence we can consider categories enriched over , i.e. -categories.
A locally internal category over is given by
In the stack semantics of , a locally internal category “looks like” an ordinary locally small category.
If we also take care of the appropriate morphisms have the following:
(2b) (Robert Pare) If this indexed functor preserves pullbacks (as an indexed functor) and if it induces the given enrichment, this extension is unique (up to a canonical isomorphism). Elephant B2.2.8
Of course, does not have to be a topos. For the definition, it suffices for to have finite limits, although the notion is best-behaved when is locally cartesian closed (for instance, in that case the codomain fibration of is an example).
J. Penon, Categories localement internes, C. R. Acad. Sci. Paris 278 (1974) A1577-1580
Locally internal categories, Appendix in: P. Johnstone, Topos theory, 1977
Chapter B2.2 of Sketches of an Elephant