This mission begins immediately after The Atonement, as you face-off with John once and for all.
Remember that these planes are two seaters- you'll have to hop up on the wing to enter the cockpit, rather than the turret gun in back.
The main difference between this plane and, say, the Carmina, is that it has rockets. Just get John in your sights and light him up.
Taking out John's plane is a checkpoint moment, so if it looks like you might be able to down him but will die trying, go for it. Either way, you'll have to track him down on the ground afterwards. Pleasingly, there's no overblown boss-fight, a single bullet to the head will settle matters between you.
This launches a cutscene in which you take the bunker key, and hear John's last words. It also unlocks the Affirmation for use, and as fighter planes go it's arguably the best in the game.