This is how you do something when you are being pedantic: in a fussy, rule-loving, detail-heavy way. “He pedantically corrected every comma” means he went through the text and treated every tiny mark as a big problem. The colour is the same as pedantic — almost always unkind, or at least tired. It is formal. In speech, people more often say “in a pedantic way” than “pedantically”. It does not mean “like a teacher” and it does not mean pedagogically (in a way that helps people learn). A teacher can explain something pedagogically without being pedantic at all.

學究氣地、吹毛求疵地