  We consider a large class of nonlinear FPKEs with coefficients of
Nemytskii-type depending explicitly on time and space, for which it is known
that there exists a sufficiently Sobolev-regular distributional solution u in
L^1 and L^\infty. We show that there exists a unique strong solution to the
associated McKean-Vlasov SDE with time marginal law densities u. In particular,
every weak solution of this equation with time marginal law densities u can be
written as a functional of the driving Brownian motion. Moreover, plugging any
Brownian motion into this very functional produces a weak solution with time
marginal law densities u.
