  In this manuscript we develop a version of Szemer\'edi's regularity lemma
that is suitable for analyzing multicolorings of complete graphs and directed
graphs. In this, we follow the proof of Alon, Fischer, Krivelevich and M.
Szegedy [Combinatorica, 20(4) (2000), 451--476] who prove a similar result for
graphs.
  The purpose is to extend classical results on dense hereditary properties,
such as the speed of the property or edit distance, to the above-mentioned
combinatorial objects.
