On this page of our game guide to Batman: Arkham Knight we provide a detailed description of the option duel with additional boss - Professor Pyg. Learn how to defeat his vanguards and then how to finish him at the end.
The battle with Professor Pyg takes place during the final phase of The Perfect Crime side quest during which you investigate the mutilated corpses in various parts of Gotham City. You will be fighting the mentioned enemy once you reach the operating room in Pretty Dolls Parlor.
First you must fight patients of the mad doctor. It might be tricky as regular attacks cannot eliminate them for good. The only way to permanently get rid of an opponent is to perform ground takedown on him after he falls to the ground. You must remember that performing the finisher takes some time and can be cancelled by other enemies. Because of that, you should activate the takedown only when you're standing near a lonely knocked down enemy.
After you eliminate the first group of patients, professor Pyg and a new group of his minions will join the fight. During this battle, you should ignore the Pyg for most of the time and react only if he throws cleavers at you. Once that happens, press the counter key/button in order to block his attacks and throw back one of the cleavers. Focus only on performing takedowns on new patients during the rest of the battles. As earlier, try to pick only single enemies that are away from the main group, it will reduce the probability of failing the finisher.
Take interest in professor Pyg only after you get rid of all the patients. Wait until he throws another cleaver at Batman and perform the counter to throw one of the weapons back to its owner. Once Pyg is temporarily stunned, run towards the operating table highlighted in blue (picture above) and perform the takedown on the enemy - it will automatically end the battle.