If you say that something duly happened or was done, you mean that it was expected to happen or was requested, and it did happen or it was done.
Westcott appealed to Waite for an apology, which he duly received.
韦斯科特如愿以偿地得到了韦斯特的道歉。
If something is duly done, it is done in the correct way.
He is a duly elected president of the country and we're going to be giving him all the support we can.
他是这个国家正式选出的总统,我们将给他我们所能给的一切支持。