Probability of arising of simple graph in configuration model I am studying a configuration model building $d$-regular graphs and reading the following article:
The expansion of random regular graphs by David Ellis.
I am stuck on the following step:

Each simple labelled $d$-regular graph on $n$ vertices has the same probability of arising. Indeed, it is easy to see that a simple d-regular graph on $[n]$ arises from precisely $(d!)^n$ of the matchings.

Where does this $(d!)^n$ come from?
Consider a simple $d$-regular graph. For each vertex, assign numbers from $1$ to $d$ to the edges adjacent to it. There are $(d!)^n$ ways to do so. Each such assignment corresponds to a particular perfect matching in the configuration model, and vice versa.