Castelnuovo-Mumford regularity of representations of certain product categories

We show in this paper that representations of a finite product of categories satisfying certain combinatorial conditions have finite Castelnuovo-Mumford regularity if and only if they are presented in finite degrees, and hence the category consisting of them is abelian. These results apply to examples such as the categories ${\mathrm{F}\mathrm{I}}^m$ and ${\mathrm{F}\mathrm{I}}_G^m$.