A/C evaporator thermostat switch
If your AC system is working OK, this switch prevents freezing of the evap coil by opening the circuit to the compressor clutch coil when temp drops to ~40F. Thus the compressor should cycle on and off with the air temp coming out of the evaporator not dropping below about 40F. As you move the temp control lever to a warmer setting, a cable to the switch increases the setting and causes the compressor to cycle on and off at a higher temperature.
If the switch is out of the evaporator, I you could test it in an ice water bath to ensure it is open circuit as the temp drops below 40F.
Hope that helps
Jim