In Greek mythology, Caunus (Ancient Greek Καῦνος) is the twin brother of Byblis, son of Miletus (by Cyane, Tragasia, or Eidothea in variants). When Byblis confesses incestuous love, he flees through Caria and Lycia; tradition makes him founder of the Carian city Caunus (Kaunos). Distinct from the historical city alone without the Milesian twin myth.
Adapted from Wikipedia, "Byblis" · CC BY-SA 4.0
I.Twin of Byblis
Ovid's Metamorphoses makes Caunus and Byblis twins of Miletus and Cyane daughter of Maeander. Byblis sends a love letter; Caunus flees in horror. She pursues him through Asia Minor until she wastes into a weeping spring. Parthenius and Conon know a version in which Caunus himself feels the forbidden desire and leaves before he yields; Byblis hangs herself after the search.
Antoninus Liberalis has Byblis refuse other suitors and leap from a cliff, saved and nymphified by hamadryads. Nonnus reverses pursuit: Caunus sings after Byblis. Foundation legend ties the fugitive to the Carian city that bears his name.
II.Homonym note
The atlas id caunus-miletus is the Milesian twin and eponym. The archaeological city Kaunos is the place named for him, not a second character node.