If $c$ is a coupling between two measures, $c= \mu^1\, \, t \, \, \mu^2$, ($t$ is the symbol of binary operator of the coupling (I can't find a more proper symbol here)). The measures are both defined on a product space $\Omega = \prod_{s \in S} \Omega_s$. how is it justifiable writing that,
$(\mu^1 \circ \pi^{-1}_s)\, \, t \, \, (\mu^2 \circ \pi_s^{-1} ) \, \, \, = \, \, \, (\mu^1 \, t \, \, \mu^2)\, \, \circ \, \pi_s^{-1}\, \,\,$ tha means that the projection of the coupling is the coupling of the projections?
P.S. the coupling is not necessarily the product between the measures.