We revisit the composition of neutral hydrogen, i.e. atomic (HI) and molecular (H2), in the local Universe, with focus on the H2-to-HI mass ratio R in various galaxies and the H2-mass function (MF). First, we derive the H2-MF from the CO-luminosity function (LF) of the FCRAO Extragalactic CO Survey, using a variable CO-to-H2 conversion fitted to nearby observations. This implies a universal H2 density Omega(H2)=(6.9±2.7)*10^-5/h and Omega(H2)/Omega(HI)=0.26±0.11, which could, however, suffer from incompleteness of the CO-sample. Therefore, a second approach derives the H2-MF from a HI-sample with well described completeness (HIPASS). This premisses an estimation of R, for which we introduce a set of phenomenological models based on a sample of 245 galaxies drawn from the literature. Our best model in terms of statistical scatter describes R as a function of galaxy type and cold gas mass. This model leads to an H2-MF that matches the one derived from the CO-LF, thus supporting that model and settling completeness uncertainties.