At the end of "The Custom House", in the novelThe Scarlet Letter, the author ends his narrative hoping that
It may be...that the great-grandchildren of the present race may sometimes think kindly of the scribbler of bygone days, when the antiquary of days to come, among the sites memorable in the town's history, shall point out the locality of THE TOWN PUMP.
This is basically a hope that the author (Hawthorne), as well as the story, can remain present in the minds of future generations. It is also a hope that the future generations that will live in that town are willing to preserve their history and the many stories that have survived as society keeps moving forward, and as it changes. Forgetting and neglecting to remember the origins of how things happen render us vulnerable to repeat the mistakes of the past.
Hawthorne also writes this closing paragraph seemingly as a reaction to a previous statement which, with melancholy, he declares how he, too, has been at one point rejected by his own people; these are the reasons behind his wanting to deflect his literary career.
I am a citizen of somewhere else. My good townspeople will not much regret me, for...there has never been, for me, the genial atmosphere which a literary man requires in order to ripen the best harvest of his mind.
Therefore we can already notice a sense of connection between the narrator and the character of Hester; the two are people of specific intellectual attainments, and their personal philosophies-aside from the untimely expression of them- have set them aside to the point of being shun by their own.