Rather than being a Resident Of The TV World, Teddie is one of the many Grim Reapers spread around the world. Specifically, he's a five year old Reaper employeed to Inaba. Initally only being referred to as "Reaper," Yosuke quickly names him Teddie when the two get into a conversation with Nanako. Liking the name, Teddie starts going by it in general, and stops wearing his red cloak all the time, instead wearing overalls. He dons the name Reaper and the cloak again when reaping.
Teddie is usually invisible to everyone, but Persona users can bypass this, as discovered when Yosuke started talking to him when they were on their way to rescue Yukiko. After Teddie confronted him, he began living at the house, later making himself visible to everyone when Yosuke tells his parents about Teddie, despite the fact that Reapers aren't supposed to do that... or bond with humans.
Teddie can sense when someone is going to be reaped soon, even if it's via the TV World (this makes him feel more ill than a regular reap), but of course can't do anything about it but reap the person when their time comes. The only reason he could tell the others about future TV World victims is because that is another world - Reaper laws don't really apply there. He could sense Nanako's future fate, but kept ignoring it because she's so young - surely he has to be sensing it wrong.
Of course, he wasn't, and has to reap Nanako while she's in the hospital, informing Dojima about it and the fact he's a Reaper before doing so. Afterwards, he decides to try something that is not allowed for Reapers - reviving a person. Nanako's back, but she's less peppy and cheery, she knows she died (due to her young age she doesn't quite realize what happened though), and she can see Teddie whether or not he wants her too. Dojima also recalls that he saw Teddie the night his wife died.
Teddie was also affected by Nanako's revival - namely, he's missing his eyes. He can still see, yes, but people don't want to look him in the face anymore, so he is paying his price. The Investigation Team and Nanako don't seem to mind much, though he'll likely have to tell his parents the truth because there's really no other way to give a explaination for what happened.
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The deadline for Naoto comes and goes, and the Investigation Team fails to save her. Teddie refuses to reap her as well, so her ghost begins to haunt the party, but most commonly Teddie since he's the only one who can see her. He's heavily suppressing how distraught he is despite not really getting to know Naoto that well until after her death, and keeps stealing her hat to wear. The others do not know they are being haunted or that Naoto was never reaped.
The longer Naoto goes without being reaped, the more fucked up her ghost becomes, becoming foggy, loosing her legs, etc. The worst of it so far is her eyes vanishing and constantly pouring pitch black sludge, which can make contact with Teddie, but nothing else. Naoto also starts acting not like herself, and is begging Teddie to just let go and reap her. Teddie still won't.
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