  For Hamiltonian field theories on polysymplectic manifolds with a symmetry
group action and a momentum map, we explore the redundancy in a set of
necessary conditions that has appeared in the literature, for a generalized
version of the Marsden-Weinstein symmetry reduction theorem. Next, we prove a
necessary and sufficient condition for polycosymplectic reduction. We relate
polycosymplectic reduction in a one-to-one way to the reduction of an
associated larger polysymplectic manifold. Throughout the paper, we provide
examples and discuss special cases.
