Let COMP (respectively TYCH) by the category of compact Hausdorff spaces (respectively Tychonoff spaces). The notion of a normal endofunctor F: TYCH\(\to TYCH\) is introduced, and it is shown that every normal endofunctor G: COMP\(\to COMP\) in the sense of E. V. Shchepin [Usp. Mat. Nauk 36, No.3(219), 3-62 (1981; Zbl 0463.54009)] can be extended to a normal endofunctor F: TYCH\(\to TYCH\) which, in a certain sense, is unique.