Depending what stands for various things can be called -category. E.g. if is a category then we talk about categories over , for them see overcategory.
This entry is rather about another notion of -category introduced in
which is similar to the notion of -category of Rosenberg. It is called -category as it is suitable context for a generalization of a separable functor.
A little info is found in Toen: Homotopical and higher categorical structures in algebraic geometry. File Toen web unpubl hab.pdf. S-cats are closely related to Segal cats.
nLab page on S-category