All 5 Uses

correspond
in
Washington Square, by Henry James
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He supposed she was in correspondence with Morris Townsend; but he held his peace about it, for he never saw the young man's letters, and Catherine's own missives were always given to the courier to post.†
Chpt 24 *

correspondence = communication by written letters or messages
I haven't the least idea; I am not in secret correspondence with him!†
Chpt 30 *

correspondence = done from afar
She would have liked to have a lover, and to correspond with him under an assumed name in letters left at a shop; I am bound to say that her imagination never carried the intimacy farther than this.†
Chpt 2
It is highly probable you have been corresponding with him for the last three weeks—perhaps even seeing him.†
Chpt 19 *
And he answered her in words that corresponded to the look: "It may not prevent me, either, from losing the society of still another."†
Chpt 19

Definitions:

(1) (correspond as in: corresponding time period)
connect or fit together by being equivalent, proportionate, or matched

(Two things are equivalent if they have the same or very similar value, purpose, or result.)
(2) (correspond as in: corresponding by email)
communicate -- typically by writing letters or email

A corresponding secretary is an officer of an organization who is responsible for managing the organization's correspondence and keeping a record of it.
(3) (correspondence as in: a correspondence course)
done from afar

For example, a corresponding member or a correspondence course.

This sense of corresponding arose because people who lived in distant cities and could not be present for meetings, could communicate by sending written communications.
(4) (correspondent as in: foreign correspondent of the paper)
a reporter or other representative -- typically from a foreign country or with a particular expertise
(5) (meaning too rare to warrant focus)