Does Forgefire Cape's AoE trigger again if Azir switches into combat wearing it?

Attack and defend triggers happen only the first time a unit gains that designation each combat. Because an equipment's ability at the bottom of the card functionally belongs to whichever unit currently has the gear attached, Azir gaining the relevant designation for the first time in that combat — right after you move Forgefire Cape onto him — means he also gets a Forgefire Cape trigger.