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To be honest, I would have wanted to see Batgirl as being Robin for a while maybe before becoming Batgirl or directly herself again.
I am not a huge fan of Carrie Kelley at all, she is already from an alternate universe. Robins are normally four to me, Dick Grayson, Jason Todd, Tim Drake and Damien Wayne. Damien is even open to be argued, yet if we don't count Barbara they are four.
I don't think there should be in the cinematic universe. I would love to see Dick Grayson if I had a choice. Batgirl as herself could be like Robin, since she would need a training from Bruce.
Why exactly? Since Batman and Robin are set, established characters and they've always been men why would you change him to a woman? Batgirl is a woman, I don't understand why a 70 year old classic male character would be changed to another gender.
Sure, come up with new female characters if you want, I don't understand why you'd change a classic character.