If you activate the ability granted by Deadeye Navigator, the creature will be exiled, the pair will immediately be broken, and then the card will be returned to the battlefield. Deadeye Navigator’s soulbond ability triggers when that card enters the battlefield and the pair can then be reunited.

Once Deadeye Navigator or the creature it’s paired with is exiled, the other creature will no longer have the activated ability. However, you can activate the ability of one creature in response to activating the ability of the other creature.