All 3 Uses of

correspond
in
The Scarlet Letter

Day after day she looked fearfully into the child's expanding nature, ever dreading to detect some dark and wild peculiarity that should correspond with the guiltiness to which she owed her being.
p. 84.0 *

correspond = connect or fit together

It might be, on this one day, that there was an expression unseen before, nor, indeed, vivid enough to be detected now; unless some preternaturally gifted observer should have first read the heart, and have afterwards sought a corresponding development in the countenance and mien.
p. 212.0

corresponding = fitting

He was heard to speak of Sir Kenelm Digby and other famous men—whose scientific attainments were esteemed hardly less than supernatural—as having been his correspondents or associates.†
p. 112.1 *

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