Disc galaxies are complex systems shaped by different stellar structures such as bulges, discs, and bars. Every galaxy structure keeps a unique memory of their past evolution, thus, understanding the mass assembly of disc galaxies can only be achieved by analysing separately all these independent footprints of past evolution. In this work, we have analysed the separated star formation main sequence (SFMS) of bulges and discs in the CALIFA sample. To this aim, we use the C2D algorithm to separate the spectra of bulges and discs using the information provided by the CALIFA integral field survey. We find that the star formation in galaxies mainly occurs in the disc component and not in bulges. Remarkably, once we only account for the disc mass, even the discs of early-type galaxies are compatible with the SFMS defined by star forming galaxies at z $\sim$ 0. Moreover, we find a strong mass depedence in the SFMS. For the bulges, only the low-mass bulges in late-type spirals ($\log{M_{\star,\rm{b}}/M_{\odot}} <$ 9) are compatible with the SFMS of z $\sim$ 0 star-forming galaxies. The fraction of discs compatible with the SFMS is also a function of the disc (or total) stellar mass. Our results remark the importance of separating the spectral properties of different galaxy structures to provide a whole picture of the galaxy mass assembly throughout cosmic time.