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The intensity mapping of the [CII] 158um line redshifted to the sub-mm window is a promising probe of the z>4 star formation and its spatial distribution into the large-scale structure. To prepare the first-generation experiments (e.g., CONCERTO), we need realistic simulations of the sub-mm extragalactic sky in spectroscopy. We present a new version of the SIDES simulation including the main sub-mm lines around 1mm (CO, [CII], [CI]). This approach successfully reproduces the observed line luminosity functions. We then use our simulation to generate CONCERTO-like cubes (125-305GHz) and forecast the power spectra of the fluctuations caused by the various astrophysical components at those frequencies. Depending on our assumptions on the relation between star formation rate and [CII] luminosity, and the star formation history, our predictions of the z~6 [CII] power spectrum vary by two orders of magnitude. This highlights how uncertain the predictions are and how important future measurements will be to improve our understanding of this early epoch. SIDES can reproduce the CO shot noise recently measured at ~100 GHz by the mmIME experiment. Finally, we compare the contribution of the different astrophysical components at various redshift to the power spectra. The continuum is by far the brightest, by a factor of 3 to 100 depending on the frequency. At 300GHz, the CO foreground power spectrum is higher than the [CII] one for our base scenario. At lower frequency, the contrast between [CII] and extragalactic foregrounds is even worse. Masking the known galaxies from deep surveys should allow to reduce the foregrounds to 20% of the [CII] power spectrum up to z~6.5. However, this masking method will not be sufficient at higher redshifts. The code and the products of our simulation are released publicly and can be used for both intensity mapping experiments and sub-mm continuum and line surveys.