  We continue the investigation of abstract Uryson operators in vector
lattices. Using the recently proved Up-and-down theorem for order bounded,
orthogonally additive operators, we consider the domination problem for
AM-compact abstract Uryson operators. We obtain the Dodds-Fremlin type theorem
and prove that for an AM- compact positive abstract Uryson operator T from a
Banach lattice E to a order continuous Banach lattice F, every abstract Uryson
operator S from E to F included between 0 and T is also AM-compact.
