Using three correlated band approaches, namely the conventional band approach plus on-site Coulomb repulsion U, the modified Becke-Johnson functional, and hybrid functional, we have investigated inverse-Heusler ferrimagnets Cr2CoL (L =Al, Ga, In). These approaches commonly indicate that the Cr2CoAl synthesized recently is a precise compensated half-metal (CHM), whereas Cr2CoGa and Cr2CoIn are ferrimagnets with a small moment. This is also confirmed by the fixed spin moment approach. Analysis of the Bader charge decomposition and the radial charge densities indicates that this contrast is due to chemical differences among the L ions. Additionally, in Cr2CoAl, changing the volume by +/- 5% or the ratio of c/a by +/- 2% does not alter the CHM state, suggesting that this state is robust even under application of moderate pressure or strain. Considering the observed high Curie temperature of 750 K, our results suggest that Cr2CoAl is a promising candidate for robust high TC CHMs. Furthermore, the electronic structure of the CHM Cr2CoAl is discussed.