  We calculate the three-loop matching coefficient $C_{HH}$, required for a
consistent description of Higgs boson pair production in gluon fusion through
next-to-next-to-leading order QCD in the heavy top quark approximation. We also
compute the $gg \to HH$ amplitude in $m_t \to \infty$ approximation in the full
theory and show its consistency with an earlier computation in heavy-top
effective theory.
