Pure semisimple finitely accessible categories and Herzog's criterion

Let C be a finitely accessible category with products, and assume that its symmetric category s(C) is also finitely accessible and pure semisimple. We study necessary and sufficient conditions in both categories for C (and hence s(C)) to be of locally finite representation type. In particular, we obtain a generalization of Herzog's criterion for finite representation type of left pure semisimple and right artinian rings. As an application, we prove that a left pure semisimple ring R with enough idempotents which has a self-duality is of locally finite representation type if and only if it is left locally finite.