In particular, when hunting a mouse, the cat is thinking about what the mouse is thinking. The "if I hide here, the mouse won't be able to see me when it leaves its mouse hole."
That establishing of a model of another animal's mind in their own mind is likely a key part of intelligence.
Yes -- this is literally "theory of mind", as I understand it.
But the really striking thing is that this sort of stuff you're talking about does not typically count towards theory of mind among scientists.
I mean they would make the counterargument that the cat catching the mouse might simply be learning through reinforcement that relying on their hearing is effectively more successful than relying on both sight and hearing, and that being unable to see the mouse and focussing on the sound leads to better strategies; it's possible the cat does not need to understand that they are invisible to the mouse.
And it is certainly possible for a cat to catch a mouse through long observation of its behaviours without engaging in theory of mind.
But then you watch cats play with each other (or the hilarious videos of cats sneaking up on observers and then freezing) and it's hard to believe they don't understand this.
This is a very, very interesting video.
There are other interpretations (that it is for example mum/dad's favourite toy too for example). But it is worthy of discussion.