I just got one from the Mystery Box. Does it turn your pet into a Lumi or what does it do?

According to Subetalodge:

Not sure what you mean by Lumi though (but I mean, no it's just a stat booster it seems) :x

You need an , Nostalgic Instructions, and to complete Vanya's Quest for Thimble & Co. items if you want a Nostalgic pet.

Thank you for your help, Ladies.

You're welcome. :)