We present a backward approach for the interpretation of the evolution of the near-IR and the far-IR luminosity functions (LFs) across the redshift range 0\lt z\lt 3. In our method, late-type galaxies are treated by means of a parametric phenomenological method based on PEP/HerMES data up to z ∼ 4, whereas spheroids are described by means of a physically motivated backward model. The spectral evolution of spheroids is modeled by means of a single-mass model, associated with a present-day elliptical with a K-band luminosity comparable to the break of the local early-type LF. The formation of proto-spheroids is assumed to occurr across the redshift range 1≤slant z≤slant 5. The key parameter is represented by the redshift {{z}0.5} at which half of all proto-spheroids are already formed. For this parameter, a statistical study indicates values between {{z}0.5}=1.5 and {{z}0.5}=3. We assume {{z}0.5}∼ 2 as the fiducial value and show that this assumption allows us to describe accourately the redshift distributions and the source counts. By assuming {{z}0.5}∼ 2 at the far-IR flux limit of the PEP-COSMOS survey, the PEP-selected sources observed at z\gt 2 can be explained as progenitors of local spheroids caught during their formation. We also test the effects of mass downsizing by dividing the spheroids into three populations of different present-day stellar masses. The results obtained in this case confirm the validity of our approach, i.e., that the bulk of proto-spheroids can be modeled by means of a single model that describes the evolution of galaxies at the break of the present-day early-type K-band LF.
