The circumstances of encountering the boss: Firefly is the fifth main boss that appears during the game's campaign mode. You are going to fight this boss after you defused three bombs on the Gotham Pioneers Bridge, right after the completion of the Disarm the bomb at the north pillar mission.
Start the fight with Firefly by pressing the counter-attack key/button, thanks to which Batman will use the claw and make it over to the fight arena in a quite unconventional way (only then, will the boss' health bar appear).
Right after the fight's main part starts, you should spend a few moments learning Firefly's attacks. The boss is predominantly using his flame thrower and to avoid taking damage, you need remain in constant motion. Performance of leaps should not be necessary, although it can be helpful at critical stages and let you avoid losing health.
At moments in between Firefly's attacks, you can try throwing the glue grenades at him, and it is best to use the key/trigger combination that allows you to throw one quickly (i.e. by skipping manual aiming). After several attempts, one of the grenades should hit the boss and immobilize him for a moment (the screenshot).
Start now throwing regular batarangs. Just like earlier, rely on the key/trigger combination that allows you to throw them quickly. It is also a good idea to have an upgrade that makes the batarangs-throwing process more efficient. Still, you can do without it.
Continue throwing batarangs up o the moment when the game zooms in on the boss and suggest that he has been stunned temporarily. Quickly, use the combination of keys/buttons for quick use of the claw. After the boss is captured, start tapping the key/button that appears on the screens to pull him towards yourself. Batman will then land a strong blow without your participation.
From the moment of performing the first combination (glue grenade - batarangs - claw) Firefly will start to use another attack, which is the bombardment of the area. In this case, running around the bridge is not enough anymore and, according to the game's suggestion, you should also perform leaps to the sides. For the whole time, you should be on a lookout for a good opportunity to hit the boss with another glue grenade, to be able to perform the actions described above once again. .
The next phase of the fight should start at the moment when the Firefly loses around a half of his healthbar. Start escaping, using the hook. Walk over to the left over the metal element of the bridge. You will reach another interactive point shown in the screenshot (you need to use again the hook).
Another part of the escape assumes maneuvering between car wrecks and, do not forget to rely on leaps to the sides at the moments at which Firefly starts bombarding the area. Jump onto the blue vehicle and get towards the truck's trailer, under which you need to perform a slide.
You can now continue the actions that have been described at the beginning on this page. Keep running around the bridge, avoid Firefly's attacks and wait for opportunities to throw him the glue grenades. Each time that you notice that the Firefly was stunned , pepper him with batarangs, and then, use the claw on him (do not worry if you are not fast enough to shoot the claw, because the batarangs themselves will also deal damage to Firefly).
After the boss' healthbar is nearly depleted, he will start bombardment of the area. After you avoid the first wave of the charges, try to stand, more or less, in the middle of the arena, because this is going to be the only safe place. Wait for the moment in which Batman will be able to catch the Firefly with the hook (screen).
This is not the end, because during the last phase of the fight, you will have to press quickly the counter-attack button, in two different places. This will prevent the boss from smashing Batman against an obstacle.