An unlikely protagonist slinks around in “The Cat,” which follows So-yeon, a pet groomer who suffers from claustrophobia stemming from a childhood experience. One day, one of So-yeon’s customers dies on the elevator, leaving her cat Bi-dan behind. The mystery of who killed the woman on the elevator is unknown, and the cat is the only witness. As the police begin their investigation, they ask So-yeon to take care of Bi-dan. But something strange happens to So-yeon from the day she brings Bi-dan home. So-yeon starts having nightmares in which she sees a little girl who keeps coming back to haunt her dreams. Afraid and unsure of what this might mean, So-yeon starts investigating the mystery of the cat and the death of its owner when her closest friend, who has also recently gotten a cat, dies.