Let k be a commutative Noetherian ring and Cāā be a locally finite k-linear category equipped with a self-embedding functor of degree 1. We show under a moderate condition that finitely generated torsion representations of Cāā are super finitely presented (that is, they have projective resolutions each term of which is finitely generated). In the situation that these self-embedding functors are genetic functors, we give upper bounds for homological degrees of finitely generated torsion modules. These results apply to quite a few categories recently appearing in representation stability theory. In particular, when k is a field of characteristic 0, we obtain another upper bound for homological degrees of finitely generated FI-modules.