The phrase "another duty of" is correct and usable in written English.
You can use it when introducing an additional responsibility or task that someone is expected to perform.
Example: "Another duty of the project manager is to ensure that all team members are meeting their deadlines."
Alternatives: "an additional responsibility of" or "a further obligation of".
On disciplining auditors, another duty of the new board, a chairman friendly to the industry might ensure that nothing much changes.
Mr. Dumpty was assisted by a human editor named Martin Gardner, who prepared "activity features" and wrote a monthly short story about the adventures of the child egg, Humpty Dumpty Jr. Another duty of Mr. Gardner's was to write a monthly poem of moral advice from Humpty Sr. to Humpty Jr.
Another duty of the host agent, which is the main focus of this work, is to detect whether the host is overloaded or underloaded.
Another important duty of The Fabulous One is to keep everyone happy and stress free by bringing the laughs.
One must owe another a duty of care before he or she may be held liable for injuries he or she causes.
I would like to answer that question with another: what is the duty of a friend in such a situation?
To the Sikh ideals of his predecessor, Guru Hargobind thus added another: the right and duty of the Sikhs to defend their faith by the sword if necessary.
They do not turn on the right-holder's power over the duty of another, so they do not share the will theory's difficulty with unwaivable rights.
Patriotism is not but another extension of the duty of concern for others; it is a special concern for my country because it is my country, for my compatriots because they are my compatriots.
(DJN VIII.3.8) Those who have injured another certainly owe a duty of reparation, as well as assurances about future behavior, but they are under no obligation to submit to punishment (DJN VIII.3.4; see end of 4.1.3 above), which is imposed for purely consequentialist reasons.
By the love of our country, I do not mean loyalty to our King, for that is a duty of another nature, and a man may be very loyal, in the common sense of the word, without one grain of public good in his heart.
