Context. Classical novae are thermonuclear explosions that take place in the envelopes of accreting white dwarfs in stellar binary systems. The material transferred onto the white dwarf piles up under degenerate conditions, driving a thermonuclear runaway. In those outbursts, about 10-7 - 10-3 Msun, enriched in CNO and, sometimes, other intermediate-mass elements (e.g., Ne, Na, Mg, or Al, for ONe novae) are ejected into the interstellar medium. The large concentrations of metals spectroscopically inferred in the nova ejecta reveal that the (solar-like) material transferred from the secondary mixes with the outermost layers of the underlying white dwarf. Aims. Most theoretical models of nova outbursts reported to date yield, on average, outflows characterized by O > C, from which only oxidized condensates (e.g, O-rich grains) would be expected, in principle. Methods. To specifically address whether CO novae can actually produce C-rich dust, six different hydrodynamic nova models have been evolved, from accretion to the expansion and ejection stages, with different choices for the composition of the substrate with which the solar-like accreted material mixes. Updated chemical profiles inside the H-exhausted core have been used, based on stellar evolution calculations for a progenitor of 8 Msun through H and He-burning phases. Results. We show that these profiles lead to C-rich ejecta after the nova outburst. This extends the possible contribution of novae to the inventory of presolar grains identified in meteorites, particularly in a number of carbonaceous phases (i.e., nanodiamonds, silicon carbides and graphites).