Which terms are used for the digraph of the edges of a digraph? For any digraph $G = (V,E)$, consider the digraph $H = (E,F)$ with $F = \\{ee' \in E \times E\ |\ \exists u,v,w \in V: e = uv \wedge e' = vw\\}$, that is, the digraph $H$ whose nodes are the edges of the digraph $G$ and whose edges connect any two edges of $G$ which, together, form a directed path. Which term is used in the literature for the digraph $H$ wrt. the digraph $G$?

For undirected graphs this is called the line graph. In the link there is a short section on the _line digraph_.