Really loved the apartment - it was very clean, had good heating (important for someone who gets cold easily!), very cosy, had comfortable beds, and was in good location. The internet was good, too - four of us could use it at the same time and had no problems. Also, Francesco was a very responsive host, answering my messages prior to our trip almost immediately.
He and his wife greeted us, explained everything we need to know about the flat, gave us tips about getting around and even recommended places to eat. We ended up going to one of Francesco' recommendations, Ai Balestrari (Francesco also had a 10% discount card for it), and we really enjoyed it there. The restaurant had really nice atmosphere, nice service, tasty food and deserts, and really nice wine.
Only one thing to note for potential tenants is that shower cabin was quite small - we were four slim girls and still were hitting the cabin door with our elbows, so I would imagine the shower cabin might be a bit too small for somebody who is bulkier or taller.
Francesco also offers a pick-up from airport/from flat for an additional fee, so if you are someone with a lot of luggage/prefer to skip the hassle of travel on public transport/want to get to the flat/airport faster, it is something to consider. We, however, ended up not choosing the pick-up and got to and from the flat just fine. We used a mix of bus and metro.
All in all, thank you Francesco for having us at his indeed very "Sweet Vatican Apartment"!