  We consider the fully-coupled McKean-Vlasov equation with multi-time-scale
potentials, and all the coefficients depend on the distributions of both the
slow component and the fast motion. By studying the smoothness of the solution
of the non-linear Poisson equation on Wasserstein space, we derive the
asymptotic limit as well as the quantitative error estimate of the convergence
for the slow process. Extra homogenized drift term containing derivative in the
measure argument of the solution of the Poisson equation appears in the limit,
which seems to be new and is unique for systems involving the fast
distribution.
