Her mate makes a cocky display (6)

I believe the answer is:

peahen

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'display' is the definition.
Although both the answer and definition are singular nouns, I cannot see how one could define the other.

I don't understand how the rest of the clue works.

Can you help me to learn more?

(Other definitions for peahen that I've seen before include "Mrs Peacock" , "Female of a proud-tailed cock" , "Female peacock" , "Bird, gorgeous-tailed mate" , "Bird with a showy partner" .)

I've seen this clue in the Evening Standard.

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