This is correct. But there is even more involved. Before an ATP can instruct/check in "air transportation service" he/she must have reach a certain level of experience in the aircraft (usually at least 500 hours of line PIC time), undergo in-house training, including observation rides, and then in the case of becoming a check airman, must be observed by a FAA inspector while performing the type of checking the ATP will do. Lastly, they must receive a LOA from the FAA regarding the type of instructing/checking permitted. The average ATP on the line may not instruct in any legal way without that LOA, and then only in their company's FAA approved training program to their company's pilots. This would include type ratings for company aircraft, and if necessary, initial ATP certificates.

With the exception noted above, only FAA certified flight instructors may give flight instruction to students persuing FAA certificates and ratings.