  We use high-resolution hydrodynamical simulation to test the difference of
halo properties in cold dark matter (CDM) and a self-interacting dark matter
(SIDM) scenario with a constant cross-section of
$\sigma^\text{T}/m_{\chi}=1\;\text{cm}^{2}\text{g}^{-1}$. We find that the
interplay between dark matter self-interaction and baryonic physics induces a
complex evolution of the halo properties, which depends on the halo mass and
morphological type, as well as on the halo mass accretion history. While high
mass haloes, selected as analogues of early-type galaxies, show cored profiles
in the SIDM run, systems of intermediate mass and with a significant disk
component can develop a profile that is similar or cuspier than in CDM. The
final properties of SIDM haloes - measured at z=0.2 - correlate with the halo
concentration and formation time, suggesting that the differences between
different systems are due to the fact that we are observing the impact
self-interaction. We also search for signatures of self-interacting dark matter
in the lensing signal of the main haloes and hints of potential differences in
the distribution of Einstein radii, which suggests that future wide-field
survey might be able to distinguish between CDM and SIDM models on this basis.
Finally, we find that the subhalo abundances are not altered in the adopted
SIDM model with respect to CDM.
