Sefer M’Deshen Baysecha explains the dispute between Sarah and Hagar with the following story:
A wealthy man once had to travel to Prague for business, and he hired a wagon driver to take him there. On the way, night fell and they stopped at an inn to sleep.
In the early morning hours, while the wealthy man was still asleep, the wagon driver put on the rich man’s clothes and claimed to be him. He ordered the rich man to take the reigns of the horses, saying that he was the wagon driver.
When they got to Prague, the wealthy man, who was now dressed like a wagon driver, ran to the Rov of the city, the Nodah B’Yehuda, and told him what the real wagon driver had done to him. The wagon driver, on the other hand, insisted that he was really the boss and the other man was lying.
The Nodah B’Yehuda told them to come back in the morning. He told his gabbai that when the two men came back, he should instruct them to wait in the waiting room, and to make them stay there for a while. The Rov left the men sitting there for hours before he suddenly stuck his head into the waiting room and said, “Wagon driver come here!”
Immediately, the real wagon driver stood up without thinking. He realized his mistake and tried to sit back down but by that point, it was obvious who he really was.
Similarly, Hagar claimed that she was no longer Sarah’s servant. She said that since she was married to Avrohom, she no longer had to work for her. Sarah countered that Hagar still belonged to her, and Avrohom ruled that Sarah was correct. Therefore, Hagar fled. The angel then came to her and called out, “Hagar, Sarah’s maidservant”. When she immediately said “here I am”, it became obvious that she really still was Sarah’s servant.