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We use high-resolution hydrodynamical simulation to test the difference ofhalo 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 theinterplay between dark matter self-interaction and baryonic physics induces acomplex evolution of the halo properties, which depends on the halo mass andmorphological type, as well as on the halo mass accretion history. While highmass haloes, selected as analogues of early-type galaxies, show cored profilesin the SIDM run, systems of intermediate mass and with a significant diskcomponent can develop a profile that is similar or cuspier than in CDM. Thefinal properties of SIDM haloes - measured at z=0.2 - correlate with the haloconcentration and formation time, suggesting that the differences betweendifferent systems are due to the fact that we are observing the impactself-interaction. We also search for signatures of self-interacting dark matterin the lensing signal of the main haloes and hints of potential differences inthe distribution of Einstein radii, which suggests that future wide-fieldsurvey 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 adoptedSIDM model with respect to CDM.

The interplay of Self-Interacting Dark Matter and baryons in shaping the halo evolution