  We study the problem of defining maps on link Floer homology induced by
unoriented link cobordisms. We provide a natural notion of link cobordism,
disoriented link cobordism, which tracks the motion of index zero and index
three critical points. Then we construct a map on unoriented link Floer
homology associated to a disoriented link cobordism. Furthermore, we give a
comparison with Oszv\'ath-Stipsicz-Szab\'o's and Manolescu's constructions of
link cobordism maps for an unoriented band move.
