  We derive the CO luminosity function (LF) for different rotational
transitions (i.e. (1-0), (3-2), (5-4)) starting from the Herschel LF by
Gruppioni et al. and using appropriate $L'_{\rm CO} - L_{\rm IR}$ conversions
for different galaxy classes. Our predicted LFs fit the data so far available
at $z\approx0$ and $2$. We compare our results with those obtained by
semi-analytical models (SAMs): while we find a good agreement over the whole
range of luminosities at $z\approx0$, at $z\approx1$ and $z\approx2$ the
tension between our LFs and SAMs in the faint and bright ends increases. We
finally discuss the contribution of luminous AGN
($L_{X}>10^{44}\,\rm{erg\,s^{-1}}$) to the bright end of the CO LF concluding
that they are too rare to reproduce the actual CO luminosity function at
$z\approx2$.
