In the 2001 movie, Monsters, Inc., we see that there are monsters that walk into the rooms of little kids, scare the kids, and come back out, and this creates energy for Monstropolis.
And we have a little kid in Toy Story and Toy Story 2: Andy Davis. This theory is asking if Andy has a monster.
Let's start off: The worlds of Monsters, Inc. and Toy Story are connected because in Boo's room we see a Jessie doll (not the Jessie doll, a Jessie doll).
The monsters are traveling into the world of Toy Story to scare kids.
Monsters go into the kids' rooms through their closet doors. Luckily, we get a very good shot of Andy's door here:
Andy's door is white, with five rectangular panels. So all we have to do is find a door in Monsters, Inc. that is white with five rectangular panels.
Some people say they have found it. I haven't.
But in the prequel to Monsters, Inc. (Monsters University), there is a college named Monsters University. Pixar created a website named MonstersUniversity.com, which they laid out as a real college website (which just goes to show how brilliant Pixar is).
Once you go on the website, you can watch a video titled 'A Message from the Dean.' You can see it here. At about the 0:36 mark, we see this:
If you look on the far, far left between the two scream tubes -- what's that we see? It looks kind of like a door. Kind of like a white door. In fact, maybe a white door with five rectangular panels.
THAT'S ANDY'S DOOR! AHHH!
But that just proves Andy's door exists in the Monsters, Inc. world. It doesn't say anything about who goes through that door, but let's take a look at that picture again, instead let's focus on the wallpaper this time.
It's blue with clouds on it.
And at the beginning of the day when all the monsters are warming up, getting ready to scare, look at one of the wallpapers Randall is warming up against:
Blue with clouds on it.
RANDALL IS ANDY'S MONSTER.
There's one more thing that might prove that Randall is Andy's monster. In Toy Story 3, look at what we see on Andy's door.
That was actually intended to be an Easter egg for a movie called Newt, a Pixar movie that was going to be released in 2012, but was canceled and replaced with Brave.
It says "Newt Xing," and this could mean something else... Andy put a Newt Xing sticker on his closet door, because Randall most resembles a newt. This may mean that Andy no longer believes it is a monster, but a newt and a big shadow. He put the sticker on the closet door as a reminder of that.
Not only that, but he has a post on this website about this theory here.