When Remus first came back to Hogwarts, he snuck into Filch's office and looked through his drawers for a very specific object: the Marauder's Map. He figured he might find it there since Filch confiscated it from him and his friends years ago, but it seems some other students have discovered and made use of it since then.
He forgot about it for a little while. After all, as a professor, he wouldn't have much use for it. Until one night during Christmas Break. He was patrolling the hallways when he heard a noise, and then a shush, and then footsteps.
"Homenum Revelio" Remus whispered, and the charm led him to the students. Unbeknownst to him, these two students were able to track his location, and could see him coming. Unbeknownst to them, he was able to track them too. They could run, but they can't hide.
Finally, after a chase that lasted several minutes, the Weasley twins were cornered.
"See? It's just a parchment professor."
But it was more than just a parchment, and Remus knew that. He recognized it. The Marauder's Map. Any other professor could've been fooled by it, but not Remus. He created this map.
"Then you wouldn't mind if I kept it." He played along. The twin seemed disappointed.
He went back to his office, sat down in his chair, put the map on his desk, and for a little while just stared at it. He can't believe that after all this time, he finally got it back.
"I solemnly swear that I am up to no good," Remus pointed his wand to the map, and watched as it activated.
He looked through the castle. He checked on Harry in the Slytherin dorms. He was there, and so was Draco Malfoy. He then checked the faculty room, Snape was there, up to who knows what. He then checked the forest. Sirius. Remus wondered if the twins noticed him or not.
"Wait a minute…" Remus had an idea. If the map reveals Sirius in his Animagus form, then it should also reveal Peter, unless Sirius is lying.
He flipped through the map, looking for the Gryffindor tower. And there he was: Peter Pettigrew.
The Bastard! Remus thought he was dead. He thought he was betrayed. He thought… he thought… he thought he was his friends.
But if what Sirius says to be true is true, if Sirius isn't lying about this…
What if there's an explanation to all this? What if Peter is innocent? And Sirius is lying? But then… why would Peter hide for twelve years? Especially after Sirius was sent to Azkaban? And Voldemort dead?
He should go to Dumbledore with this… Wait, no! He can't! He'd be exposing his three friends as unregistered Animagi.
Well, I mean… James is dead. What are they gonna do to him?
They're already after Sirius…
And if Peter is a traitor, he's going to Azkaban for way worse…
Still, he has no evidence to support his claim, except for a map created by a bunch of Hogwarts students in their fifth-year. He needs to know for sure that Sirius isn't lying, and he would need to have more evidence to back his claim when he goes to Dumbledore.