The Kondo effect is a many-body phenomenon, allowing insight into the electronic and atomistic structure of magnetic adsorbates on metal surfaces. Its chemical control is intriguing because it deepens such insight, but the underlying mechanisms are only partly understood. We study the effect of increasing the number of CO ligands attached to a cobalt adatom on copper(001), which correlates with an increase in the Kondo temperature T K experimentally [Wahl et al., Phys. Rev. Lett. 95, 166601 (2005)], by solving an Anderson impurity model parametrized by the density functional theory. Our results suggest that the orbital responsible for the Kondo effect is d x 2 - y 2 for the tetracarbonyl and its combination with d z 2 for the dicarbonyl. The molecular structures depend considerably on the approximate exchange-correlation functional, which may be related to the known difficulty of describing CO binding to metal surfaces. These structural variations strongly affect the Kondo properties, which is not only a concern for predictive studies but also of interest for detecting mechanical deformations and for understanding the effect of tip-adsorbate interactions in the scanning tunneling microscope. Still, by constraining the tetracarbonyl to C 4 v symmetry, as suggested by experimental data, we find structures compatible with the experimental trend for T K (employing BLYP-D3+U). This is not possible for the tricarbonyl despite the range of computational parameters scanned. For the tetra- and dicarbonyl, the increased T K correlates with a larger hybridization function at the Fermi level, which we trace back to an increased interaction of the Co 3 d orbitals with the ligands.