I don't think there's enough context here to give you a proper answer. But, watchful means to observe something in anticipation of having to act if it becomes necessary. So a doctor might be engaged in watchful waiting of a patient: checking on a the progress of a patient's illness before deciding to operate or medicate.

Hey Kimura, to be clear, "vigilante" and "vigilant" are different words. "Vigilante" (noun) is a person who illegally enforces the law according to their own perception. "Vigilant" (adjective) describes someone who watches for danger. A police officer is vigilant because he/she watches carefully for danger in order to protect citizens. Regarding this sentence, "watchful" would loosely translate to "carefully paying attention to something", in this case, the alignment of shared values. Please let me know if I need to clarify anything!

You could rephrase it this way "The purpose of humanities and art subjects (for example, history, languages, theatre and music) is to help society make sense of the difference between what individual people feel is the right way to live and what society in general says is the correct way to live." Watchful is in there to make it seem that the arts have an important role in this. Hope this helps!