As soon as you get to the rooftop the real boss battle will begin. Mundus will try to attack you with his fists. Dodge out of the way.
When you dodge him, his hand will come off and reveal a red crystal inside it. Use Angel lift to quickly get near it then start chopping away and deal some damage.
His right and left arm have their respective HPs, you have to damage both in order to proceed further. Also his slamming attack can be dodged easily by performing a double jump as long as you're away from his targeted area. You'll know where he's targeting cause a red mark will appear on the floor.
His second attack is a lot easier to dodge, he'll just slam his hand down three times. Avoid it by rolling to the opposite side at the right time.
If he vomits lava on the ground get to the right side and use Angel lift to get to a different rooftop and continue the battle.
If he uses the wave attack then just hop over it and Angel lift towards his hand and deal some more damage.
After destroying one of his hands, Mundus will lean towards you and a Demon pull mark will appear on one of his eyes. Use Demon pull on it and a short cut scene will occur.
Get Mundus to attack you again and keep on hitting the arm he has left until it breaks. When it does break he'll expose his other eye and you'll be able to pull it out with Demon pull.
He'll also use his lava vomit a bit more now that he doesn't have any hands. Just use the boost ring to get to the other side and avoid the damage.
Mundus will also have a new attack now. Meteors will fall from the sky. They're pretty easy to avoid though just keep rolling towards one side and you should escape without any damage.
After he uses the meteor attack Mundus will place his head near the rooftop. Use Angel lift to instantly close in on his mouth and start attacking it.
Once you fully deplete his health Dante will jump inside and kill Mundus' human shell ending the battle.