A mechanic was notified of a cooling issue on a refrigerator trailer in the fleet. He had just brought the trailer in from the yard and parked in the work bay of the trailer shop to begin the troubleshooting process. While performing the tension checks, the thermostat on the trailer triggered the engine to start. A belt pulled the mechanic's hand into a pulley and his fingers were caught at a pinch point. He sustained an amputation to the left middle finger at the first joint, and an amputation to the left index finger at the base of the fingernail.