The film is set in the Qing Dynasty during the 43rd year (1779) of the reign of the Qianlong Emperor. Li Mu Bai (Chow Yun-fat) is an accomplished Wudang swordsman. Long ago, his master was murdered by Jade Fox (Cheng Pei-pei), a woman who sought to learn Wudang skills. Mu Bai is also a good friend of Yu Shu Lien (Michelle Yeoh), a female warrior. Mu Bai and Shu Lien have developed feelings for each other, but they have never acknowledged or acted on them due to Yu's deceased fiance Meng Sizhao, who was also Mu Bai's best friend. Mu Bai, intending to give up his warrior life, asks Shu Lien to transport his sword, also referred to as the Green Destiny, to the city of Beijing, as a gift for their friend Sir Te (Sihung Lung). At Sir Te's estate, Shu Lien meets Jen (Zhang Ziyi), the daughter of Governor Yu (Li Fazeng), a visiting Manchu aristocrat. Jen, destined for an arranged marriage and yearning for adventure, seems envious of Shu Lien's warrior lifestyle.
One evening, a masked thief sneaks into Sir Te's estate and steals the sword. Mu Bai and Shu Lien trace the theft to Governor Yu's compound and learn that Jade Fox has been posing as Jen's governess for many years. Mu Bai makes the acquaintance of Inspector Tsai (Wang Deming), a police investigator from the provinces, and his daughter May (Li Li), who have come to Peking in pursuit of Fox. Fox challenges the pair and Sir Te's servant Master Bo (Gao Xi'an) to a showdown that night. Following a protracted battle, the group is on the verge of defeat when Mu Bai arrives and outmaneuvers Fox. Before Mu Bai can kill Fox, the masked thief reappears and partners with Fox to fight. Fox resumes the fight and kills Tsai before fleeing with the thief (who is revealed to be Fox's protĂŠgĂŠ, Jen). After seeing Jen fight Mu Bai, Fox realizes Jen had been secretly studying the Wudang manual and has surpassed her in combative skills.
At night, a desert bandit named Lo (Chang Chen) breaks into Jen's bedroom and asks her to leave with him. A flashback reveals that in the past, when Governor Yu and his family were traveling in the western deserts, Lo and his bandits had raided Jen's caravan and Lo had stolen her comb. She chased after him, following him to his desert cave seemingly in a quest to get her comb back. However, the pair soon fell passionately in love. Lo eventually convinced Jen to return to her family, though not before telling her a legend of a man who jumped off a cliff to make his wishes come true. Because the man's heart was pure, he did not die. Lo came to Peking to persuade Jen not to go through with her arranged marriage. However, Jen refuses to leave with him. Later, Lo interrupts Jen's wedding procession, begging her to come away with him. Nearby, Shu Lien and Mu Bai convince Lo to wait for Jen at Mount Wudang, where he will be safe from Jen's family, who are furious with him. Jen runs away from her husband on the wedding night before the marriage could be consummated. Disguised in male clothing, she is accosted at an inn by a large group of warriors; armed with the Green Destiny and her own superior combat skills, she emerges victorious.
Jen visits Shu Lien, who tells her that Lo is waiting for her at Mount Wudang. After an angry dispute, the two women engage in a duel. Shu Lien is the superior fighter, but Jen wields the Green Destiny: the sword destroys each weapon that Shu Lien wields, until Shu Lien finally manages to defeat Jen with a broken sword. When Shu Lien shows mercy and lowers the sword, Jen injures Shu Lien's arm. Mu Bai arrives and pursues Jen into a bamboo forest. Following a duel where Mu Bai regains possession of the Green Destiny, he decides to throw the sword over a waterfall. In pursuit, Jen dives into an adjoining river to retrieve the sword and is then rescued by Fox. Fox puts Jen into a drugged sleep and places her in a cavern; Mu Bai and Shu Lien discover her there. Fox suddenly reappears and attacks the others with poisoned darts. Mu Bai blocks the needles with his sword and avenges his master's death by mortally wounding Fox, only to realize that one of the darts hit him in the neck. Fox dies, confessing that her goal had been to kill Jen because she was furious that Jen had hid the secrets of Wudang's far superior fighting techniques from her.
As Jen exits to gather up an antidote for the poisoned dart, Mu Bai prepares to die. With his last breaths, he finally confesses his romantic affections for Shu Lien. He dies in her arms as Jen returns, too late to save him. The Green Destiny is returned to Sir Te. Jen later goes to Mount Wudang and spends one last night with Lo. The next morning, Lo finds Jen standing on a balcony overlooking the edge of the mountain. In an echo of the legend that they spoke about in the desert, she asks him to make a wish. He complies and wishes for them to be together again; back in the desert. Jen then suddenly leaps over the side of the mountain.

Question: What is Jen's relationship to Governor Yu?
Answer: Daughter