  To fathom the mechanism of high-temperature ($T_{\rm c}$) superconductivity,
the dynamical vertex approximation (D$\Gamma$A) is evoked for the
two-dimensional repulsive Hubbard model. After showing that our results well
reproduce the cuprate phase diagram with a reasonable $T_{\rm c}$ and dome
structure, we keep track of the scattering processes that primarily affect
$T_{\rm c}$. We find that local particle-particle diagrams significantly screen
the bare interaction at low frequencies, which in turn suppresses
antiferromagnetic spin fluctuations and hence the pairing interaction. Thus we
identify dynamical vertex corrections as one of the main oppressors of $T_{\rm
c}$, which may provide a hint toward higher $T_{\rm c}$'s.
