On page 4 of 1 , you find:
[...] a multiplex network can be represented as a collection of graphs $$\mathcal{G}=\{G^{(\ell)}=(V_n,E^{(\ell)})\}_{\ell \in V_L}$$ where $V_n=\{1,\ldots,n\}$ is the set of nodes, $V_l=\{1,\ldots,L\}$ s the set of layers and $E^{(\ell)}\subset V_n\times V_n$ is the set of edges on layer $\ell$.
These structures are also sometimes called multi-layer graphs. Note that in the above formulation it is assumed that the nodes on each layer are the same (this models the bijective edges you are referring to).
1 Node and layer eigenvector centralities for multiplex networks. F Tudisco, F Arrigo, A Gautier - SIAM Journal on Applied Mathematics, 2018 (arXiv)