The adsorption of H, CO2, HCOO, O and CO on copper monolayers and submonolayers supported on hexagonal WC(0001) surfaces has been investigated. Calculations have been performed using density functional theory with the Perdew-Burke-Ernzerhof exchange correlation functional and D2 van der Waals corrections. In addition, dipole corrections were also included. The catalytic properties of supported Cu on both carbon- and metal-terminated WC(0001) surfaces were explored. On carbon-terminated WC(0001) surfaces, Cu tends to be oxidized, while on the metallic terminated surface, it gains charge. The results indicate that all studied Cu/WC(0001) surfaces bind all adsorbates stronger than the extended Cu(111). For CO, the binding energy is so large in some cases (1.6-2.2 eV) that it could potentially lead to catalyst deactivation. Nevertheless, surfaces with an adsorbed Cu monolayer, CuML, are less prone to this deactivation, since there are not WC surface atoms; and thus, the contribution of strong CO adsorption from the support does not play a role. Energy barriers for HCOO formation, relative to direct dissociation barriers of CO2, indicate that a hydrogen-assisted reduction path is more likely to occur on Cu/WC(0001) materials, with CuML/metallic termination being the most active system for this reaction path. On the other hand, CO2 adsorption on CuML surfaces is slightly weaker on a C-terminated surface than on a metal-terminated surface, although both surfaces have similar dissociation barriers. This fact together with the weaker CO adsorption on CuML/C-terminated WC(0001) than on metal-terminated WC(0001) suggests that the former system may be a better catalyst for CO2 reduction, due to the lower surface poisoning by the CO2 dissociation products. Possible deactivation of Cu/WC(0001) materials may be prevented by the introduction of hydrogen into the system, thus promoting the formation of HCOO and avoiding CO and O formation.