  In this paper we give estimates for the proton--proton cross sections into
pairs of quarkonium states $J/\psi$, $\psi(2S)$, $\Upsilon(1S)$ and
$\Upsilon(2S)$ at the scheduled AFTER@LHC energy of $115$GeV. The estimates are
based on the intrinsic heavy flavor mechanism which is observable for high
values of $x_F$, a range outside the dominance of single parton and double
parton scattering.
