The flat itself is all right.
What was really infuriating was the check-in procedure:
The host had more than 1 month to ask us to verify our ID or send copies if necessary (I don' t think it really is and my ID has been verified here, also supported my a decent number of host feedback)I
Hence, requesting a proof of our ID 2 mins before arrival, refusing to provide the check-in code before, was really annoying!
I wonder how you can expect people from a foreign people to be always online in Italy?
Just by chance I had the EU WiFi subscription, otherwise we would be screwed!
After such an experience, I cannot really recommend these hosts!