Electronic and transport properties of CuGaTe$_2$, a hole-doped ternary copper based chalcopyrite type semiconductor, are studied using calculations within the Density Functional Theory and solving the Boltzmann transport equation within the constant relaxation time approximation. The electronic bandstructures are calculated by means of the full-potential linear augmented plane wave method, using the Tran-Blaha modified Becke-Johnson potential. The calculated band gap of 1.23 eV is in agreement with the experimental value of 1.2 eV. The carrier concentration- and temperature dependent thermoelectric properties of CuGaTe$_2$ are derived, and a figure of merit of $zT= 1.69$ is obtained at 950 K for a hole concentration of $3.7cdot10^{19}$ cm$^{-3}$, in agreement with a recent experimental finding of $zT= 1.4$, confirming that CuGaTe$_2$ is a promising material for high temperature thermoelectric applications. The good thermoelectric performance of p-type CuGaTe$_2$ is associated with anisotropic transport from a combination of heavy and light bands. Also for CuSbS$_2$ (chalcostibite) a better performance is obtained for p-type than for n-type doping. The variation of the thermopower as a function of temperature and concentration suggests that CuSbS$_2$ will be a good thermoelectric material at low temperatures, similarly to the isostructural CuBiS$_2$ compound.
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