Adam Clarke on 3 John 1:4

To hear that my children - From this it has been inferred that Caius was one of St. John's converts, and consequently not the Corinthian Caius, who was converted, most probably, by St. Paul. But the apostle might use the term children here as implying those who were immediately under his pastoral care, and, being an old man, he had a right to use such terms in addressing his juniors both in age and grace; and there is much both of propriety and dignity in the appellation coming from such a person.