Locally Self-Injective Property of FI^(m)

We consider the locally self-injective property of the product FI^(m) of category FI 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 FI^(m)-module over a field of characteristic O is injective,and the Serre quotient of the category of finitely generated FI^(m)-modules by the category of finitely generated torsion FI^(m)-modules is equivalent to the category of finite-dimensional FI^(m)-modules.