It's - no, that's not... no.
"Babysittin'" is when you pay someone to look after a baby when you goin' somewhere and can't take care of it for a couple days.
Don't ask why it's called that, 'cause I dunno.
We use that for other things, too - "housesittin'", "petsittin'", whatever.

[He waves the whole thing off with his hand, before he looks back at the little dog in Nen's arms.
Mind if he leans in to pet the li'l thing too? 'Cause he's gonna.]

So... whatcha think?