On the road to Padua in the middle of the afternoon, Petruchio looks up at the sun and says the "moon" looks beautiful. Kate corrects him, but Petruchio says it's the moon or whatever else he says it is. Otherwise, nobody's going to the wedding, capice?
Hortensio whispers to Kate to play along with Petruchio, so Kate says "OK, fine, it's the moon." "No," says Petruchio, "it's the sun." "OK," says Kate, "it's the sun or the moon, whatever you want it to be."
They encounter an old man (the real Vincentio) on the road to Padua and Petruchio pretends the guy is a young girl. Kate plays along this time and agrees with whatever Petruchio says.
When the party learns that Vincentio is looking for his son Lucentio, Petruchio says he may as well be his father, too, since Lucentio is set to marry Kate's sister. They're one big happy family.
Vincentio is surprised, but they assure him it's true and agree to take him to Lucentio's house.
Petruchio, Kate, and Vincentio head off, but Hortensio stays behind. He says he's looking forward to his own marriage because he now knows how to deal with the widow if she gets mouthy and tries to act up.