  We perform direct thermovoltage measurements in a single-electron transistor,
using on-chip local thermometers, both in the linear and non-linear regimes.
Using a model which accounts for co-tunneling, we find excellent agreement with
the experimental data with no free parameters even when the temperature
difference is larger than the average temperature (far-from-linear regime).
This allows us to confirm the sensitivity of the thermovoltage on co-tunneling
and to find that in the non-linear regime the temperature of the metallic
island is a crucial parameter. Surprisingly, the metallic island tends to
overheat even at zero net charge current, resulting in a reduction of the
thermovoltage.
