  In this paper, we prove that discrete Morse functions on digraphs are flat
Witten-Morse functions and Witten complexes of transitive digraphs approach to
Morse complexes. We construct a chain complex consisting of the formal linear
combinations of paths which are not only critical paths of the transitive
closure but also allowed elementary paths of the digraph, and prove that the
homology of the new chain complex is isomorphic to the path homology. On the
basis of the above results, we give the Morse inequalities on digraphs.
