In the theory of superconductivity, the phonon-mediated electron-electron scattering leads to an effective interaction, the BCS hamiltonian,$$\hat{H}_{\rm BCS}=\sum\limits_{\vec k,\sigma}\epsilon_{\vec k}c^\dagger_{\vec k,\sigma}c_{\vec k, \sigma}-\kappa^2\sum\limits_{\vec k, \vec k^\prime}c^\dagger_{\vec k,\uparrow} c^\dagger_{-\vec k,\downarrow} c_{-\vec {k'},\uparrow} c_{\vec {k'}, \downarrow}$$ which represents an attractive interaction between pairs of electrons (or Cooper pairs) rather than between two electrons.
So my question is, what does phonon exchange do? Does it result in an attractive interaction between two electrons of equal and opposite momenta and spins OR does it cause an attractive interaction between two Cooper pairs $|\vec k,\uparrow;-\vec k,\downarrow\rangle$ and $|\vec k',\uparrow;-\vec k',\downarrow\rangle$ OR both?