  We extend the theory of probability graphons, continuum representations of
edge-decorated graphs arising in graph limits theory, to the 'right
convergence' point of view. First of all, we generalise the notions of overlay
functionals and quotient sets to the case of probability graphons. Furthermore,
we characterise the convergence of probability graphons in terms of these
global quantities. In particular, we show the equivalence of these two notions
of convergence with the unlabelled cut-metric convergence (and thus also with
the homomorphism densities convergence and the subgraph sampling convergence).
In other words, we prove the equivalence of the 'left convergence' and the
'right convergence' views on probability graphons convergence, generalising the
corresponding result for (real-valued) graphons (the classical continuum
representation for simple graphs).
