  $\beta$ and $\alpha$ relaxation processes are dynamical scaling regimes of
glassy systems occurring on two separate time scales which both diverge as the
glass state is approached. We study here the crossover scaling from $\beta$- to
$\alpha$- relaxation in the cooperative facilitation scenario (CFS) and show
that it is quantitatively described, with no adjustable parameter, by the
leading order asymptotic formulas for scaling predicted by the mode-coupling
theory (MCT). These results establish: (i) the mutual universality of the MCT
and CFS, and (ii) the existence of a purely dynamic realization of MCT which is
distinct from the well established random-first order transition scenario for
disordered systems. Some implications of the emerging kinetic-static duality
are discussed.
