Jim Gordon might initially be in disbelief. But then, considering all the times Batman has helped him and the city, he could come to accept it. He may also start to see Bruce Wayne in a whole new light, realizing that his playboy persona was just a cover for his nocturnal activities as Batman. He might even become more protective of Bruce's secret identity, as he knows how important Batman is for Gotham.
The theme of sacrifice also comes up. Both characters are willing to sacrifice a lot for the safety of Gotham. In some fanfictions, they have to make tough choices that show their commitment to the greater good. For example, Gordon might risk his career to protect Batman's identity at times.
Batman is often portrayed as the brooding, dark vigilante. He's highly intelligent, with a strong sense of justice. He's dedicated to cleaning up Gotham but is also a loner at heart. Jim Gordon, on the other hand, is the honest cop. He's morally upright and willing to work with Batman despite the fact that Batman operates outside the law. He's a family man and is seen as the bridge between the law - abiding citizens and Batman.
One popular Batman Jim Gordon fanfiction is 'Gotham's Guardians'. It delves deep into their partnership, showing how they rely on each other in the fight against crime in Gotham. The story has great character development for both of them.
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