New York city's cheese supplier (7)

I believe the answer is:

buffalo

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'new york city's' is the definition.
(I've seen this before)

I cannot understand how the rest of the clue works.

(Other definitions for buffalo that I've seen before include "Colloquialism for the N American bison" , "Wild ox - New York city" , "NY port; --- Bill (Cody)" , "Animal whose milk made mozzarella initially" , "part of New York state?" .)

I've seen this clue in The Times.

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