Batwoman could secretly be introducing the Arrowverse's version of Professor Pyg as the brother of villain Alice. Kate Kane has finally solved the mystery of her sister's disappearance 15 years ago. Beth was caught up in a car accident triggered by the Joker, and her unconscious form was swept down the Gotham River. She was found by Dr. August Cartwright, who imprisoned her in order to provide company for his son, Jonathan. August Cartwright's harsh upbringing appears to have driven both Beth and Jonathan insane, and Beth has taken up the name "Alice" after Lewis Carroll's classic.
Jonathan Cartwright is an odd fit for Batwoman. He's being treated as an important part of Gotham's mythology, but that frankly makes no sense at all. The Cartwrights have absolutely no comic book equivalent, and at first glance it's impossible to match Jonathan with any classic Batman villains. Given Batwoman is liberally exploiting Batman lore, that seems pretty strange. The most likely solution is that Cartwright is a proto-version of some comic book Bat-foe, one whose true background has never been revealed in the comics.
One major possibility is that Jonathan Cartwright is Professor Pyg. So far, there's actually a surprising amount of circumstantial evidence supporting the idea that he could become the Arrowverse's take on the Batman villain. He's a strange, psychologically maladjusted individual, and he seems attached to dolls. Jonathan's facial disfigurements have led him to become practiced at surgery, and he likes to take the skin from bodies, transforming it into masks that he stitches on his face. Several shots have focused on these masks floating in tubs of water - and they've looked just like the Dollotron masks Pyg makes in the comics.
Created by Grant Morrison and Andy Kubert in 2009, Professor Pyg was written to be one of the craziest Batman villains of all time. The Dark Knight's rogues' gallery is populated by a lot of insane foes, who spend most of their lives being sent to Arkham Asylum and then breaking out again, but Professor Pyg made most of them look normal. He was introduced as the head of an insane circus group, Le Cirque d'Etrange, but he's since become a solo threat. Professor Pyg is best known for transforming people into Dollotrons; he captures people and subjects them to a mysterious and horrific experiment, involving everything from ritual mutilation to genital manipulation, culminating in his stitching a doll's mask over their faces.
This wouldn't be the first time a version of Professor Pyg has been adapted for the small screen; a take on the character appeared in Gotham, with a very different origin story. But that was an original concept, created for that specific TV show, and there's no reason Batwoman's, or any other version, should follow it. Meanwhile, it would be oddly appropriate for Professor Pyg to make his Arrowverse debut in a Gotham without a Batman; that was the case in the comics as well, because he first appeared at a time when Dick Grayson had taken over the mantle of the Bat.