We present a new set of cosmological zoom-in simulations of a MW-like galaxy which for the first time include elastic velocity-dependent self interacting dark matter (SIDM) and IllustrisTNG physics. With these simulations we investigate the interaction between SIDM and baryons and its effects on the galaxy evolution process. We also introduce a novel set of modified DMO simulations which can reasonably replicate the effects of fully realized hydrodynamics on the DM halo while simplifying the analysis and lowering the computational cost. We find that baryons change the thermal structure of the central region of the halo to a greater extent than the SIDM scatterings for MW-like galaxies. Additionally, we find that the new thermal structure of the MW-like halo causes SIDM to create cuspier central densities rather than cores because the SIDM scatterings remove the thermal support by transferring heat away from the center of the galaxy. We find that this effect, caused by baryon contraction, begins to affect galaxies with a stellar mass of
and increases in strength to the MW-mass scale. This implies that any
simulations used to constrain the SIDM cross sections for galaxies with stellar
masses between and at least M will require baryons to
make accurate predictions.