To do something correctly is to to it accurately: to do it right. A correctly spelled word is spelled right.

When something is correct, it is right or accurate. The correct answer to a math problem is the right answer, so if you answered a math problem right, you answered it correctly. A correctly used semi-colon will make English teachers happy. If you quote a movie correctly, you get the quote exactly right. If you follow the traffic laws correctly, the police won't bother you.

Definitions of correctly

adv in an accurate manner

“the flower had been correctly depicted by his son”
aright, right
incorrectly, wrong, wrongly
in an inaccurate manner

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