Prior to 1962 the Navy had a different numbering system than the Army/Air Force. The actual number for the Wildcat was the F4F, and yes, the Corsair was the F4U. The first letter indicated the category of aircraft (fighter, attack, etc.) The last letter was the manufacturer code (F was Grumman's, U was Vought's). The number indicated how many of each category the manufacturer had produced. So the Wildcat was the 4th fighter that Grumman had developed for the Navy and the Corsair was the 4th fighter that Vought had developed
In 1962, the DOD created the unified designation system, requiring all the services to use the same designation for a aircraft systems across the services. For example, the McDonnell Phantom II, designated F4H by the Navy and F-110 by the Air Force, became the F-4.
Joe Baugher's web site has a great explanation, I've provided a link.