During the Golden Age/sliver age of comics Batman was deputized by Commissioner Gordon, which gave Batman an advantage when he was taking down criminals in his Rogues gallery, because instead of being sent to a mental Asylum they were actually sent to prison.
during the Bronze Age post crisis on infinite Earths Batman was stripped of his deputation and made a vigilante who operated outside the law, this actually created a problem. Because when Batman arrest one of his rogues like the Joker instead of being sent to prison like he should be he sent to Arkham Asylum, and the only reason is because he was arrested by a vigilante and not an actual cop.
As an example look at the Batman animated series episode trial. District Attorney Janet Van Dorn was trying to put a stop to poison ivy for Good by getting her life in prison sentence, However when the judge gives his ruling because poison ivy was captured by vigilante known as the Batman and not an actual cop he decides to have her sent to Arkham Asylum.
Note: at this time poison ivy in DCAU wasn't a metahuman yet she was an eco-terrorist, so the DA pushing to put Ivy behind bars for life makes a lot of sense given the number of time she's tried to assassinate people for wronging Mother Nature.
another example would be in the Batman animated series episode Riddler's reform summer Gleeson is giving the news report on the penguin the quote of dialogue from the episode "Still no verdict in the trial of the Penguin though apprehended last month by Batman the jury remains unconvinced of the felon's guilt despite overwhelming" this is another instance where Batman being a full vigilante is more of a hindrance, since the jury is hesitant to even convect the villain, since we're having to look at the evidence that was collected by a vigilante not a deputized officer of the law.
now if Batman was re deputized it would eliminate a lot of the drawbacks that have resulted from Batman being a vigilante who operates outside the law. as an example more Batman's Rogues Gallery would actually be sent to Blackgate Penitentiary and not Arkham Asylum.