Panels (a) and (b): Relation between τ (a) and t₀ (b), the two parameters of our SFH model (Equation A.1), and the galaxy M_★ at z = 0, constrained by the Popesso et al. (2023) main sequence (MS) of star formation. The blue dots show the models in our library that minimise the distance to the MS at various z. The red-dashed curves show our regularisation using fifth degree polynomials. Panel (c): SFR-M_★ relations at four different redshifts. The solid curves show the MS of Popesso et al. (2023), the filled circles show the sequence emerging from our fiducial SFH library. Panel (d): Our fiducial SFH library adopted for MS galaxies, colour-coded by the galaxy M_★ at z = 0. Our models support the downsizing scenario of M_★ build-up.