Collecting for Agitha
The Female Dragonfly is one of the 24 Golden Bugs in Twilight Princess. Its male counterpart is the Male Dragonfly. This bug can be traded to Agitha for a Rupee prize.
The Female Dragonfly can be found in Zora's River, right next to Iza's Rapid Ride Shop. Once Link is standing near the door, he can use the Gale Boomerang to bring this bug to him.
The Male Dragonfly can be found in Zora's Domain.
Female Dragonfly's location in Zora's River
Male and Female Dragonfly's locations on the Wii map