Consider a typical high bypass turbofan designed for cruise at approximately mach 0.8, with a separate nozzle for the bypass air. Ideally, the bypass air would be compressed adiabatically by the intake and the fan, and then expanded adiabatically in the nozzle to ambient pressure and temperature. But there are losses in the intake, fan and nozzle, all of which generate a little bit of heat.1 I also expect some heat is absorbed from the engine core.
Moreover, according to this answer, the bypass air exiting the nozzle in cruise is typically under-expanded, which might raise the temperature at the nozzle exit some more. The aforementioned answer also states that the bypass air is exiting near the speed of sound.
As this answer explains, the core airflow can get significantly faster than the speed of sound in ambient air and stay subsonic; the speed of sound is higher in the exiting air because of its higher temperature.
This makes me wonder, what is the temperature difference between the bypass air exiting the nozzle and the ambient air? In particular, is it enough to raise the speed of sound in the exiting air a meaningful amount?
1: The kinetic energy at 300 m/s is 3002/2=45000 j/kg. The specific heat of atmosphere is around 1 kj/(kg*K), so even 10% losses would only raise the temperature 4-5K. Hence little bit.