  This series of papers aims at understanding the formation and evolution of
non-barred disc galaxies. We use the new spectro-photometric decomposition
code, C2D, to separate the spectral information of bulges and discs of a
statistically representative sample of galaxies from the CALIFA survey. Then,
we study their stellar population properties analising the
structure-independent datacubes with the Pipe3D algorithm. We find a
correlation between the bulge-to-total ($B/T$) luminosity (and mass) ratio and
galaxy stellar mass. The $B/T$ mass ratio has only a mild evolution with
redshift, but the bulge-to-disc ($B/D$) mass ratio shows a clear increase of
the disc component since redshift $z < 1$ for massive galaxies. The mass-size
relation for both bulges and discs describes an upturn at high galaxy stellar
masses (log{(M_{\star}/M_{\sun})} > 10.5). The relation holds for bulges but
not for discs when using their individual stellar masses. We find a negligible
evolution of the mass-size relation for both the most massive (log{(M_{\star
\rm ,b,d}/M_{\sun})} > 10) bulges and discs. For lower masses, discs show a
larger variation than bulges. We also find a correlation between the S\'ersic
index of bulges and both galaxy and bulge stellar mass, which does not hold for
the disc mass. Our results support an inside-out formation of nearby non-barred
galaxies, and they suggest that i) bulges formed early-on and ii) they have not
evolved much through cosmic time. However, we find that the early properties of
bulges drive the future evolution of the galaxy as a whole, and particularly
the properties of the discs that eventually form around them.
