# Graph homomorphism with a non-mapping relation

In [1] it is said that a graph homomorphism is a mapping between two graphs, that is, between their vertices, where the edges are preserved.

A mapping is a specific binary relation where any vertex in a graph has a unique image in the other graph.

Why must the binary relation be a mapping in the graph homomorphism definition ? Cannot it be simply a binary relation ?

In the image below is depicted what would happen if a binary relation $\mathcal{R}$ is not a mapping, which I find sound.