(adjective)
somehow equivalent, comparable, or accompanying in some way, having a relationship in form, nature, or function;
e.g. Each box has a corresponding lid that fits it uniquely.
(adjective)
somehow involved in communicating from a distance, usually through letters/mail, for example participating in something through this medium or made responsible for writing letters for some purpose;
e.g. He is a corresponding participant of the club since he cannot attend their meetings.
(verb)
agreeing with something or matching it very closely, being similar or equivalent to it in some aspect such as form, nature, or function;
e.g. Her handwriting has a high chance of corresponding to that in the letter we found.
(verb)
communicating with someone through the exchange of written mail/letters;
e.g. The two of them have been corresponding by writing each other letters for several years.