Left: cosmoDC2 halo M_200c masses (from the simulation) versus the redMaPPer cluster richnesses. The points are colored with the redMaPPer cluster redshift. The solid line is the best-fitted mean richness-mass relation in Eq. (16) at z = 0.4. The dashed lines represent the low mass and low richness cut used on the cosmoDC2-redMaPPer matched catalog for the fit of the fiducial scaling relation. Right: Histogram of richnesses in bins of mass, and best-fit probability distributions P(lnλ|M, z) from Eq. (15) (solid lines) for the different mass bins.