Let the adjacency matrix of an undirected bipartite graph be $A = \begin{pmatrix} 0 & B \\ B^T & 0 \end{pmatrix}$ where B is called the biadjacency matrix.

Now, by instead interpreting B as an adjacency matrix of a not necessarily bipartite (arbitrary) directed graph, we get a bijection between the bipartite undirected graph (with biadjacency matrix B) and the arbitrary directed graph (with adjacency matrix B).

Does this bijection have a name? Is it discussed somewhere in literature? Thank you!