Locally Self-Injective Property of FI m

We consider the locally self-injective property of the product [Formula: see text] of category [Formula: see text] of finite sets and injections. Explicitly, we prove that the external tensor product commutes with the coinduction functor, and hence preserves injective modules. As corollaries, every projective [Formula: see text]-module over a field of characteristic 0 is injective, and the Serre quotient of the category of finitely generated [Formula: see text]-modules by the category of finitely generated torsion [Formula: see text]-modules is equivalent to the category of finite-dimensional [Formula: see text]-modules.