Petruchio, Katherine, Hortensio, and some of Petruchio's servants are making the journey from Petruchio's house to Padua. Though it is the middle of the day, Petruchio comments on how brightly the moon shines. Katherine corrects him and Petruchio refuses to continue their journey until she says that he is right.
Petruchio demands Katherine's complete obedience, even when he is clearly wrong. This is, for him, the ideal structure of a marriage.
Katherine relents and agrees that it is the moon shining, not the sun. Petruchio immediately changes his mind and says that it is the sun and Katherine is wrong. Katherine agrees with him that the sun is shining. Hortensio comments that Petruchio has tamed Katherine and that "the field is won," (iv.5.26).
Katherine seems to be affected by Petruchio's abusive course of "taming." But has she truly changed so dramatically, or is she merely pretending?
Pleased that Katherine is obedient, Petruchio prepares to continue the journey, but just as they are setting out again, they encounter an old man. Petruchio addresses the man as a young woman, praising her beauty. He tells Katherine to embrace the young woman, and Katherine complies. She calls him a "young budding virgin, fair and fresh and sweet," (iv.5.41) but then Petruchio calls her mad and says that he is clearly an old man. Katherine apologizes.
Petruchio now tests how far Katherine's obedience will go. As it turns out, she is willing to (pretend to?) obey him even if it means acting like a complete fool.
The old man introduces himself as Vincentio and says that he is traveling to Padua to see his son Lucentio. Petruchio happily greets him as his father-in-law, since Lucentio is marrying Katherine's sister. Vincentio is confused by this and thinks that Petruchio is playing a joke on him. They all continue their journey to Padua together, with Hortensio commenting on how remarkably Petruchio tamed Katherine.
In Petruchio's view, the marriages between Katherine and him and between Bianca and Lucentio have united all of their three families. Hortensio continues to see Petruchio's dominance over Katherine as an instructive model for other men to learn from.