Batman and Batwoman’s relationship is not one that is defined by hatred and loathing. It started off like that because Batwoman operated on her own terms and did not do what Batman asked of her on several occasions in the comics, this was mostly because of how Batwoman was under the control of the DEO for some time in the N52 comics. Batman knew this, and was very much against her being used by them, but following the attack on Gotham that was orchestrated by the DEO and Batwoman, Bruce learns that she was doing this because her sister was captured by the DEO and was being manipulated into doing all of this, for the sole purpose to learn Batman’s identity; which Kate refused to give it up on many occasions, but was willing to do it for the sake of her sister.
Kate ends up incapacitating Bruce, but by this point, he is understanding of the situation Kate has been put into. They devise a plan to hand himself over to the DEO, in which they dress Kate’s father in the Batsuit, and then allow for him to be taken in. They are brought to the location, most of the Batkids show up to take down the DEO and save Kate’s sister while not having Bruce’s identity compromised or any other Batfamily’s identity. Following these events, Bruce finds Kate on a rooftop and formally acknowledges her as being Batwoman, and approves of her being in his city. Where they then both go off on patrol together, as Bruce made the offer to her.
