  Let X be a K3 surface and M a smooth and projective moduli space of stable
sheaves on X of Mukai vector v. A universal sheaf U over X x M induces an
integral transform F from the derived category D(X) of coherent sheaves on X to
that on M.
  (1) We prove that the integral transform F is faithful. F is not full if the
dimension of M is greater than 2.
  (2) We exhibit the full subcategory of D(M), consisting of objects in the
image of F, as the quotient of a category, explicitly constructed from D(X), by
a natural congruence relation defined in terms of the Mukai vector v.
  (3) Let C be a component of the moduli space of isomorphism classes of marked
irreducible holomorphic symplectic manifolds deformation equivalent to the
Hilbert scheme X^[n] of n points on a K3 surface X, n > 1. C is 21-dimensional,
while the moduli of Kahler K3 surfaces is 20-dimensional. We construct a
geometric deformation of the derived categories of K3 surfaces over a Zariski
dense open subset of C, which coincides with D(X) whenever the marked manifold
is a moduli space of sheaves on X satisfying a technical condition.
