What was found
DURING A ROUTINE PART 61 TRAINING FLIGHT, THE RIGHT ENGINE WAS INTENTIONALLY SHUTDOWN TO DEMONSTRATE INFLIGHT RESTARTS. AFTER A NORMAL SHUTDOWN AND SECURING PROCEDURE, THE ENGINE MASTER SWITCH WAS SWITCHED BACK TO THE ON IAW THE CHECKLIST. USUALLY THIS WOULD DRIVE THE PROPELLER OUT OF FEATHER, AND THE RESTART PROCEDURE COULD BE CONTINUED. THE CFI NOTICED THAT THE PROPELLER DID NOT UNFEATHER, AND THEY ATTEMPTED SEVERAL TROUBLESHOOTING PROCEDURES. IT WAS THEN DECIDED TO RETURN TO DEPARTURE UTILIZING A SINGLE-ENGINE. AFTER TROUBLESHOOTING THE PROPELLER ISSUE, IT WAS DETERMINED THAT THE GOVERNOR WAS "STICKING" AND NOT ALLOWING THE PROP TO MOVE FROM FEATHER. THE GOVERNOR WAS REPLACED WITH NEW UNIT. ALL OPS CHECKS WERE NORMAL AND THE AIRCRAFT WAS RETURNED TO SERVICE.