The purpose of this note is to clarify and complement various places in the article The classical Artin approximation theorems, published in Bull. Amer. Math. Soc. 54 (2017), no. 4, 595–633. A few reasonings in the article may have been difficult to follow or were even problematic. The author is indebted to F. Castro-Jiménez, C. Chiu, R. Ephraim, B. Lamel, S. Perlega, G. Pfister, G. Rond, M. Singer, S. Woblistin, and particularly to an anonymous referee for valuable feedback.