  Let $k$ be a commutative Noetherian ring and $\underline{\mathscr{C}}$ 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 $\underline{\mathscr{C}}$ 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 $\mathrm{FI}$-modules.
