Sayori's Parallel

Via manga updates: Okushima Sayori tripped down a flight of stairs and suddenly found herself in a parallel universe, with no idea what had happened. But she was rescued by a young man and his cat and he found a way to return her back to her own world. But instead, she finds herself traveling from universe to universe – from the medieval to the futuristic, and in every case there is a version of this man and his cat! Will Sayori ever make it home again?