  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.
