  We extend to singular schemes with Gorenstein singularities or fibered in
schemes of that kind Bondal and Orlov's criterion for an integral functor to be
fully faithful. We also contemplate a criterion for equivalence. We offer a
proof that is new even if we restrict to the smooth case. In addition, we prove
that for locally projective Gorenstein morphisms, a relative integral functor
is fully faithful if and only if its restriction to each fibre also is it.
These results imply the invertibility of the usual relative Fourier-Mukai
transform for an elliptic fibration as a direct corollary.
