satellite density profile, as confirmed by P10.

Accounting for tidal heating and disc shocking speeds up the disruption of satellite, and yields a further reduction of the mass retained by them compared with B13.

In Fig. \[fig:Fn\], we compare the cumulative number of MW satellites in terms of the circular velocity of the halo with theoretical results. The upper solid line with diamonds represents the Via Lactea subhaloes (Diemand et al. 2007). The filled squares display the set of the sum of the classical MW dwarfs and the ultra-faint-dwarfs (Simon & Geha 2007). The dashed line shows the result of our model in terms of the abundance of subhaloes in the VL2 simulations after the baryonic corrections discussed. This figure is built superimposing our results, the dotted line, to those of Simon & Geha (2007) (their Fig. 14) in the $V_c$ range 10 km/s-40 km/s. This is dictated by a) the fact that Eq. \[eq:mia\] is valid in the range $10< V_{\rm infall}< 50$ km/s, so we considered satellites with $V_{\rm infall} >10$ km/s; b) there are no halos at $z=0$ that have circular velocities over 40 km/s. This plot demonstrates clearly how applying the baryonic correction to the VL2 