With ten minutes to get back to the hospital wing, Harry and Hermione race through the castle. They hear Fudge and Snape walking towards the hospital wing but when they start in that direction themselves, they hear Peeves ahead. They duck into a classroom to avoid him and finally make it to the corridor and the hospital wing. They see Dumbledore step out of the door and run forward. He smiles at them and Harry tells him they did it. Once Dumbledore is sure that the other Hermione and Harry are gone from the hospital wing, he lets them back in. They climb into their beds and accept their chocolate without question.
It's worth noting that Dumbledore's willingness to send Harry and Hermione back in time to complete this task means that, like Lupin, he knows that this is an experience he can count on them to handle. This shows that, despite some of Dumbledore's questionable hires, his staff is capable of teaching students skills enough to accomplish tasks like this.
Harry hears a furious roar from somewhere above them and within a minute, they hear Snape's voice getting closer, shouting that Harry has something to do with Sirius's escape. Dumbledore looks calm as Snape tells Harry to reveal what he did, but Dumbledore asks Snape to think and reminds him that the door has been locked for the last ten minutes. Snape angrily leaves the ward and Fudge says that the Ministry will be in trouble once the news of Black and Buckbeak gets out. Fudge does say that the dementors will be removed from the school, as they can't be attacking students. After the adults leave, Ron moans, wakes up, and asks what happened.
Because Dumbledore sent Harry and Hermione back in time and they weren't seen, there's absolutely no evidence that they did anything under Snape's nose. This turns something that Snape used to have the upper hand on around on him, as he's no longer the one with "evidence" to support his version of events. When Fudge agrees that the dementors have to go, it indicates that at a certain point, he does recognize that the dementors have their limits.
The castle is almost deserted when Harry, Ron, and Hermione leave the hospital wing the next day. They sit by the lake and Hagrid finds them to tell them the good news about Buckbeak. Hagrid also shares that Lupin is a werewolf and just resigned. Harry runs to Lupin's office and knocks on the open door. Lupin greets him and says he saw Harry coming on the Marauder's Map. He confirms that he resigned because Snape leaked that he's a werewolf and parents will be distraught.
Now that it's public knowledge that Lupin is a werewolf, he has to deal with the consequences of this. For him, this is something normal and expected while for Harry, this kind of discrimination is new and especially painful. This then becomes another important moment in Harry's process of growing up and better understanding the world’s injustice.
Lupin asks Harry to tell him about his Patronus and confirms that James's Animagus form was a stag. As he finishes packing, Lupin gives Harry the Invisibility Cloak and the Marauder's Map. Harry asks Lupin about his earlier comment that Moony, Wormtail, Padfoot, and Prongs would've wanted to lure him out of school. Lupin explains that James would've wanted Harry to find the passages out of the castle. A knock on the door interrupts them and Dumbledore arrives to tell Lupin that his carriage is ready. Lupin bids Harry goodbye, shakes Dumbledore's hand, and leaves.
Lupin's comments about James wanting Harry to learn about the school positions the Marauder's Map as a bridge between the two generations of friends. Through the map, James is able to posthumously parent his son in a way that he otherwise could not. Through the map, he can remain young and irresponsible and join Harry as Harry grows up.
Harry flops down in Lupin's chair and says that what he did didn't make any difference since Pettigrew escaped. Dumbledore quietly says that Harry saved an innocent man and discovered the truth. This jogs Harry's memory and he tells him about Trelawney's prediction. Dumbledore seems impressed, but Harry is distraught. He says that since he stopped Sirius and Lupin from killing Pettigrew, it'll be his fault if Voldemort returns. Dumbledore tells Harry that consequences are complicated and what Harry did was noble. He points out that Pettigrew is in Harry's debt now, which could come in handy later. He assures Harry that James would've saved Pettigrew.
Dumbledore's reply to Harry's insistence that his actions made no difference suggests that as far as Dumbledore is concerned, knowing the truth on a personal level and making sure that individuals receive justice is far more important than saving the world on a grand scale. While Pettigrew may go on to help Voldemort, the lives that Harry saved through his actions are still significant.
Harry admits that he thought James conjured his Patronus across the lake. Dumbledore says that it's an easy mistake given how much Harry resembles his father, but Harry says it was a stupid mistake since he knows James is dead. Softly, Dumbledore tells Harry that he found his father inside himself through his Patronus.
Again, Dumbledore is able to use his power as a trusted mentor and teacher to encourage Harry to see that there are many ways he can connect with his father, the Marauder's Map being one and his Patronus being another.
Over the next few days, rumors fly about how Sirius and Buckbeak escaped. Malfoy is furious, while Percy spouts off about what he'll do once he gets a job at the Ministry. Harry, however, feels awful. His entire class is sad to see Lupin go, and he fixates on Trelawney's prediction. He also hates that he has to go back to the Dursleys, especially after believing for a minute that he could go live with Sirius. On the last day of term, the students get their exam results. Harry somehow makes it through Potions, but he suspects Dumbledore didn't allow Snape to fail him. In classes Snape seems to hate Harry even more.
Percy's anger in particular suggests that while he and Hermione share a love of rules and regulations, he doesn't have the critical thinking skills that Hermione developed over the course of this school year and during her trips back in time. He still thinks that the Ministry is the only way for people to receive justice and for law and order to be upheld, while Hermione knows that this isn't true.
On the train back to King's Cross, Hermione tells Harry and Ron that she's handed in her Time-Turner and dropped Muggle Studies so that she can have a normal schedule next year. Ron grouses that Hermione should've told them about the Time-Turner and then invites Harry to come and stay with him over the summer so they can go to the Quidditch World Cup. Later in the afternoon, Hermione notices a tiny gray owl outside the window. Harry grabs it out of the air and takes its letter. The letter is from Sirius. He writes that he's in hiding, admits to sending the Firebolt, and apologizes for scaring Harry in his dog form. He also encloses a note for Dumbledore giving Harry permission to visit Hogsmeade and says that Ron can keep the owl. Ron holds the owl out to Crookshanks, who purrs.
Hermione's decision to hand in the Time-Turner suggests that her purely selfish use of the device isn't worth it in the long run, while using it for the greater good—like for saving Sirius and Buckbeak—is a far more appropriate use. Even if Harry didn't think what he did was significant earlier, freeing Sirius does give Harry the one thing he's wanted all year: permission to visit Hogsmeade. Though small, this shows Harry that what he did was absolutely valuable.
Harry holds onto Sirius's letter until he arrives in King's Cross and greets Uncle Vernon. Vernon asks about the letter and refuses to sign another form, but Harry cheerfully says it's from his godfather, who's a convicted murderer and wants to know that Harry is happy. Harry feels hopeful that this will be a good summer.
Even better than getting permission to go to Hogsmeade, Harry also now has something to hold over his aunt and uncle. Now Harry has information that he can use to his advantage.