How are the edges defined in this digraph?

Question

So I am trying to understand how the number of edges are defined in this graph, I get that the number of vertices of L equals the number of edges of D, but what does the definition of E' mean for the graph? Could it be, that the new edges are defined as walks? Also, what is the meaning of "$E^2$" ?

Here's the definition: Let $L(D):= (V',E')$ be a digraph defined for the digraph $D:=(V,E)$, where L is defined in the following way:

$V':= E$

$E' := \{(e',e''\in E^2 | \exists u,v,w \in V: e' = (u,v)$ and $e''=(v,w)\}$

Answer 1

We have that $(e',e'')$ is a directed edge from $e'$ to $e''$ if in the original graph, the head of edge $e'$ intersectes the tail of $e''$. So,

becomes

.

This is a generalization of line graphs to digraphs.