Dik-dik live in hot, dry scrub and rarely drink, so every gram of water spent on cooling matters. The lining of the elongated snout is richly supplied with blood vessels, and water evaporating from that mucous membrane cools the venous blood draining from it. The animal raises the effect under heat stress by panting through the nose alone rather than open-mouthed.
That cooled venous blood is the working fluid of a selective brain-cooling system found in several bovids. At the carotid rete, a network of fine arteries bathed in the cooled venous pool, heat passes out of arterial blood before it reaches the brain. Body temperature can then be allowed to rise while brain temperature is held down, so the animal avoids the much larger water cost of cooling itself all over.
The snout has a second use. Dik-dik have prominent preorbital glands just in front of the eyes, and the black secretion is pressed onto twigs to mark territory. The two functions are separate, and the marking glands are not part of the cooling apparatus.