The size function of massive satellites from the $R_e-R_h$ and
  $M_{star}-M_h$ relations: constraining the role of environment
  In previous work we showed that a semi-empirical model in which galaxies in
host dark matter haloes are assigned stellar masses via a stellar mass-halo
mass (SMHM) relation and sizes ($R_e$) via a linear and tight $R_e-R_h$
relation, can faithfully reproduce the size function of local SDSS central
galaxies and the strong size evolution of massive galaxies (MGs, $M_{\rm
star}>10^{11.2}M_\odot$). In this third paper of the series, we focus on the
population of satellite MGs. We find that without any additional calibration
and irrespective of the exact SMHM relation, fraction of quenched galaxies or
level of stellar stripping, the same model is able to reproduce the local size
function of quiescent satellite MGs in SDSS. In addition, the same model can
reproduce the puzzling weak dependence of mean size on host halo mass for both
central and satellite galaxies. The model also matches the size function of
starforming satellite MGs, after assuming that some of them transform into
massive lenticulars in a few Gyr after infalling in the group/cluster
environment. However, the vast majority of satellite lenticulars is predicted
to form before infall. The $R_e-R_h$ appears to be fundamental to connect
galaxies and their host haloes.
