Synthetic X-ray luminosity functions using COSMIC (left) and POSYDON (right), in the case of no beaming (red dashed), and beaming with the traditional (green), or modified Eddington limit (blue). The populations include both BH and NS-ULXs, compared against observational constraints from the XLF fit of Lehmer et al. (2021, see L21 in the legend) for nearby solar-metallicity star-forming galaxies (black), and ULX demographic data (open circles) from Kovlakas et al. (2020, see K20 in the legend) using larger galaxy samples (Kovlakas et al. 2021).