larger than the target. The velocity dispersion for the observed sample on the contrary is determined by the SDSS fiber line-width, which has a fixed aperture centered on the centre of each galaxy. We argue that this difference is however smaller than the scatter we obtain in the theoretical values of the velocity dispersion $\sigma$, dominated by the intrinsic scatter in the galaxy assembly histories.

$\bullet$ Regarding the theoretical velocity dispersion, a quantity more directly comparable with data would be a luminosity-averaged (rather than mass-averaged) dispersion, but this would require an additional layer of modeling, in particular of the radial dependence of the mass-to-light ratio, that is not very well constrained observationally and represents an important source of scatter. In effect our implementation corresponds to assuming a constand mass-to-light ratio for the bulge stars; although this is not strictly true in general, it is nonetheless a fairly reasonable assumtpion for the K band, given that the bulge stars are mostly old (a few Gyrs) and that the spatial extension of the bulge is rather small compared to the disk. In addition, any differences between the mass profile and the K-band luminosity profile are smoothed out by the averaging operation (checked 