I was just approached by someone the other day who just moved to my area, and is about 2/3 of the way through his private pilot training. He asked me if I could help him finish this off. He has not flown since March, so I assume we'll have some work ahead of us. However, I think the flying side of things will be the easy part, the legality of all this is what I'm worried about.

He does not have his own airplane, so I would have to rent one from a local FBO, who has their own instructors by the way.
Would this FBO even let me do that? Do I have to tell them I'm using their aircraft for flight instruction?

Also, should he pay for my checkout in the airplane?
There's probably other questions that I haven't even thought of yet, so any insight from anyone who's done this would be very much appreciated.
If this operation is just too shady, it's really not worth putting my certs on the line to gain an extra 20 hours... Thanks in advance!