Wrong fibers of universal sheaves

Let Y be a smooth projective variety and assume M is a fine moduli space of stable sheaves on Y. Then there exists a universal sheaf U on the product Y x M. This is a family of sheaves on Y parametrized by M, such that for every m in M (given by a sheaf F), the fiber of U over m is isomorphic to F. But U can also be seen as a family of sheaves on M parametrized by Y. That is for every point y in Y we get a sheaf U_y on the wrong fiber M. We investigate the properties of these sheaves U_y on M. For example: are the sheaves U_y on M stable sheaves? This is joint work in progress with Z.Zhang.