Batman is actually the Justice League's biggest narcissist. Dear Justice League reveals the questions that children ask some of DC's greatest heroes. It also highlights how vein Batman can really be.
The Dark Knight is one of DC's most popular characters, often saving the world using his detective skills, but he also has plenty of flaws. He has some self-imposed laws that can limit his ability to end crime. He makes questionable financial decisions and has a tendency to close himself off from the people closest to him. Those aren't his only flaws.
Dear Justice League was created by Michael Northrop and Gustavo Duarte. It shows the Justice League responding to fan mail and questions received from children. When focusing on Batman, it also provides insight into just how high he is on himself. The Dark Knight is in his Batcave responding to fans - using snail mail. When Alfred informs him about electronic methods that would be much faster, Batman says he is aware of them but doesn't want to use them. Why? So he can include autographed photos of himself. Batman is definitely the slowest Leaguer in terms of response and while it would be cool to receive an autographed photo from him, it is pretty clear that he's very pleased with his photo.
This is a fantastic pose for Batman. He's perched on the rooftops of Gotham, with the bat signal in the background, ready to jump into action. However, he is the only member of the Justice League doing this. Furthermore, since he is part of a team, he could be sending a photo of the entire League, even if he still signed it and couldn't get the other members to do so. At least the caped crusader would have tried. Instead, he is perfectly happy to highlight himself and his role in Gotham as its protector.
If Batman just wanted to interact with fans, he could be responding faster. Contrarily, hand-writing a response does imply that he cares a great deal about making sure they have a personal touch. He could have designed a default message to go out to every fan or he could use just a few seconds for each response. He spends a lot of time doing detective work and patrolling the city at night, so it doesn't seem like he would have much time for doing this. Yet Batman makes it a bit of a priority and even sends something extra, even if it takes up a bit more time than an electronic response would. Even so, doing things with this approach still takes a lot of time that might be used better in other ways.
The Justice League isn't the best at keeping up with their e-mails, with the exception of the Flash. It takes up a lot of time when they already have a lot on their plates. Even though the scarlet speedster can keep up with his fans, the others aren't nearly so quick. The Justice League has to spend a lot of time-fighting and keeping an eye out for threats. Since he is the best at making deductions and finding patterns, Batman should be doing that more than anyone. From that perspective, he may be better off responding faster and foregoing the picture. Batman's fans are important, but making sure those fans live in a relatively safe world is also important - even if it means the most narcissistic Justice League member can't send cool photos of himself as a result.