Remus had wanted to be a Professor at Hogwarts and his friends were aware of it so they had themselves created a name tag of Professor R.J Lupin and pasted it on his trunk. They had made him promise that he would carry the same trunk to Hogwarts when he actually became a Professor one day.
So when in Harry's third year when the Golden Trio meets Remus on the Hogwarts Epresses, Hermione notices the name tag on his trunk and there is a mention in the book as well that the name tag seemed old as if it was peeling off slightly.
And that's because despite all that had happened, Remus kept his promise to his friends and carried the same trunk with him to Hogwarts when he was offered the job to teach Defense Against The Dark Arts in the school term of 1993.