You must apologise to cats in Takatane, Omiki, Amana and at the road until they will follow you around (takes about a minute). The cats will all appear at the road from 0:00 to 6:00 rarely to let you apologise to them all at once, however I've also listed locations below.
When you do get her to appear select "I'd like to protect you." and she'll become your partner.