the host galaxy disc.

Finally, the $\Delta v_{c}$-$v_{{\rm infall}}$ correction that we find shows a clearer trend (see the discussion in the following section). This is due to the absence, in our case, of numerical effects present, and described, in Z12[^8] are not present.

In summary, although in our model the profile flattening is calculated as in Del Popolo [*et al.*]{} (2014), here the procedure from the model of Taylor & Babul (2001) is included to follow the dynamics of satellites and their interactions with the main halo, and to take into account the mass loss during substructure evolution due to tides and tidal heating (see also Del Popolo & Gambera 1997). Moreover, inasmuch as inspired by Z12 and B13 in substructure treatment, we escaped the limitations of their SPH and SPH-based treatment with semi-analytic methods, obtaining a better $v_{c}-v_{infall}$ relation and accessing the effects of tidal heating and disc shocking. Our model employs a novel combination of parent-satellite interaction through dynamical friction, UV heating and tidal stripping to obtain satellite numbers and angular spin parameter distributions in agreement with observations.

The paper is organised as follows. In Sect. 2, we describe how the MSP and TBTF problem can be 