The first half of this duel finale is a repeat of the Bronze Tiger fight; simply cape-stun Catwoman, roll-dodging out of the way if she leaps in the air. After you've softened her up, Catwoman will opt to fight dirty, blinding Batman with a flashbang. Now the real fight begins.
While blinded, Batman is unable to move; essentially, the fight from here on out is all about timing. Wait until you see the glint of Catwoman's red goggles and hear the distinctive "shink!" sound, then counter her and quickly cape-swing to stun her, then get to punching.
In the third and final phase of the fight, multiple images of Catwoman will start spawning around you. In an effort to psych you out, some of the images will start flipping at you aggressively. <i>Don't</i> try to counter these; if you attempt to counter a Catwoman that isn't real, you'll automatically get hit. Only counter when you see the blue-lightning-bolts indicator, and you'll get the real Catwoman every time. Cape-stun, punch, and the fight is over. Congratulations, you just beat Batman: Arkham Origins Blackgate! The final cutscenes you get will change depending on what order you fought the bosses in, so keep that in mind if you start a New Game Plus.