1997 mazda protege makes funny noise when cold started

I bought a 1997 Mazda Protege 5-speed about a year ago, 179,000 miles, and it runs great but it makes a funny noise when it is started. It only does this when it is cold outside and the engine is cold, so it didn’t do it at all this summer, and it doesn’t do it in the cold if it has been warmed up and then started again. When the key is turned the engine turns over no problem, but when the key is released and it makes that small rotation back toward the driver, it makes a brief grating/honking noise. It apparently isn’t associated with any mechanical problems so far, but I am reluctant to take it on longer road trips for fear it will break down. Any ideas?