Co-universal $C^{\ast }$-algebras for product systems over finite aligned subcategories of groupoids

The product systems over left cancellative small categories are introduced and studied in this paper. We also introduce the notion of compactly aligned product systems over finite aligned left cancellative small categories and its Nica covariant representations. The existence of co-universal algebras for injective, gauge-compatible, Nica covariant representations of compactly aligned product systems over finite aligned subcategories of groupoids is proved in this paper.