Let $S$ be a countable space, $\Omega_s$ be a finite space and $\Omega_S = \prod_{s \in S} \Omega_s$ be the product space, equiped with the product topology. Let $\mu^1$ and $\mu^2$ be two probability measures defined in the product space. Let's define $\mu^1|_s $, $\mu^2|_s$ their projection on the element $s \in S$, $c = \mu^1 \, \, t \, \, \mu^2$ a coupling between the two measures, $c_s $ the same coupling, but between the projections $\mu^1|_s $ and $\mu^2|_s$, $c|_s$ the projection of the coupling $c$ on $s$.
How to prove that $ c|_s = c_s ?$