The word "Nankai" was inscribed by Mao Zedong. This is true, and it is not made up by some schools looking for words in the inscription at random.

I forgot what year it was. Some students in Nankai run a newspaper called Nankai People. At that time, someone suggested trying to write to the chairman and ask him to write an inscription. No one expected to really mention it, and wrote several copies for selection. Later, Nankai's school name always used this inscription. The word "Nankai" was written by the chairman himself, especially for Nankai University. "University" should be found in other inscriptions.

This is recorded in the history of the school.