Products and vector bundles within the category of $G$-supermanifolds

The notion of a $G$-supermanifold was introduced by the authors in order to provide reliable mathematical grounds for the approach to supergeometry expounded by De Witt and Rogers. In the present article, products within the category of $G$-supermanifolds are explicitly constructed on the lines of the general definition of products in algebraic geometry given by Grothendieck. In consequence, it becomes possible to straightforwardly introduce locally trivial superbundles.