The compatibility approach enables the description of thermoelectric (TE) effects in terms of intensive state variables. Based on the new concept of a “Thomson cooler” (using the particular case of a u = s cooler as a demonstration) the authors present practically relevant results about fully self-compatible cooler elements where the compatibility factor s is adjusted locally at any position along the element. By means of a comparison u = s cooler vs. conventional Peltier cooler with material it is shown that there is a similar self-compatibility effect for different ΔT. Therewith, the paper completes results published in Phys. Rev. B 86, 045202 (2012).