Time to read up on it, especially this part: https://www.chillingeffects.org/dmca512/faq.cgi#QID130
And: https://www.chillingeffects.org/dmca512/faq.cgi#QID922
You must notify the user and give them the opportunity to file a counter-notice against the complaint. Safe Harbor protects you from this kind of lawsuit.
Also forward a copy of the complainant's letter to chilling effects.