The insight that this conversation provides about Mr. Hooper’s character is:
It made Mr. Hooper to reveal his commitment to wearing the veil and also making him willing to accept the consequences of his decision. He also revealed his own secret sinfulness (without even naming it as that).
"The Minister's Black Veil" is known to be a short story by Nathaniel Hawthorne.
The main themes of this story by Hawthorne are hidden sin and underlying guilt. This is seen in Hooper's method of preaching that he wears his sin on his face in a literal way.
The veil actually represents a kind of hidden sin. The line from the short story suggest that: "from beneath the black veil there rolled a cloud into the sunshine, an ambiguity of sin or sorrow" (lines 335-336).
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