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In this paper we study the relative tensor product of module categories overbraided fusion categories using, in part, the notion of the relative center ofa module category. In particular we investigate the canonical tensor categorystructure and braiding inherited by the relative tensor product when modulecategories are themselves tensor/braided. As a basic example we show that thecategory of representations of the fibre product of finite groups may beexpressed in terms of the relative product and show how to multiply braidedfusion categories arising from pre-metric groups. Also, we consider relativecenters of braided fusion categories and look at the relative tensor productover Muger centers. We finish with an in-depth example hinting at acategorification of conjugacy classes for finite groups.

Relative centers and tensor products of tensor and braided fusion categories