Description of the method for fitting the metallicity gradient in the E − L_z space as a function of energy. The first column shows the distribution in the [Mg/Fe]-[Fe/H] space and the MDF for every satellite disc component, with their means (peaks) in solid (dashed) lines. The second column shows instead the final distributions in the E − L_z space of the satellite (and in situ) thin disc stars in the top (bottom) panel, colour-coded at each point with the mean metallicity. Finally, the third column shows the mean metallicity (< [Fe/H]> ) as a function of energy (E) of the satellite (and in situ) thin disc stars in the top (bottom) panel.