Binaries with a Wolf-Rayet star and a compact object (WR-COs), either a black hole (BH) or a neutron star (NS), have been proposed as possible progenitors for the binary compact object mergers (BCOs) observed with the gravitational wave (GW) detectors. In this work, we use the open-source population synthesis code SEVN to investigate the role of WR-COs as BCO progenitors. We consider an initial population of $5 \times 10^6$ binaries and we evolve it across 96 combinations of metallicities, common envelope efficiencies, core-collapse supernova models and natal kick distributions. We find that WR-COs are the progenitors of most BCOs, especially at high and intermediate metallicity. At $Z=0.02,\,{}0.014,$ and $0.0014$, more than $\gtrsim 99 \%$ of all the BCOs in our simulations evolved as WR-COs. At $Z = 0.00014$, inefficient binary-stripping lowers the fraction of BCOs with WR-CO progenitors to $\approx 83-95 \%$. Despite their key role in BCO production, only $\approx 5-30 \%$ of WR-COs end their life as BCOs. We find that Cyg X-3, the only WR-CO candidate observed in the Milky Way, is a promising BCO progenitor, especially if it hosts a BH. In our simulations, about $\approx 70-100 \%$ of the Cyg X-3 - like systems in the WR-BH configuration (BH mass $ \leq 10 \rm ~ M_\odot$) are BCO progenitors, in agreement with the literature. Future observations of WR-COs similar to Cyg X-3 may be the Rosetta stone to interpret the formation of BCOs.