Our first cat was adopted by us, we found her under the stairs to our flat where she had a little nest. We asked around if anyone owned her, put up a few flyers. No one claimed her and she looked a bit skinny and scruffy. The flat at the bottom occasionally fed her. The street was pretty much all rented flats, some sitting tenants, so I guess someone either moved on or passed away and left her.
That said, it was her choice, she lived under the stairs for several months after we sort of adopted her, she'd come in and hang about, but every night she'd want to go back to her little nest. It was more a gradual process until she finally decided that no, she'd rather stay inside.
Even then she didn't appreciate it when we moved her to the other side of London. Well, until she settled and realised she had her own garden and stuff.