  We describe an inductive machinery to prove various properties of
representations of a category equipped with a generic shift functor.
Specifically, we show that if a property (P) of representations of the category
behaves well under the generic shift functor, then all finitely generated
representations of the category have the property (P). In this way, we obtain
simple criteria for properties such as Noetherianity, finiteness of
Castelnuovo-Mumford regularity, and polynomial growth of dimension to hold.
This gives a systemetic and uniform proof of such properties for
representations of the categories $\FI_G$ and $\OI_G$ which appear in
representation stability theory.
