Warning: spoilers for the latest episode of Game of Thrones ahead!
The final season of Game of Thrones has treated us to a number of emotional reunions and tense introductions between characters we've anxiously waited to meet. Having such a large number of important characters in the same location means a lot of callbacks to the earlier seasons of Game of Thrones and also serves to remind us of exactly which pieces needed to fall into place to bring everyone to Winterfell before the great battle against the dead. One such moment occurs in the very beginning of season eight's second episode, when Jaime Lannister faces his many wrongs in a trial stacked against him.
It was expected that the former head of the Kingsguard would have to deal with the wrath of the Dragon Queen. When Daenerys brings up his crimes against her family, Sansa speaks up as well, pointing out that Jaime has wronged the Starks, too. It's only Brienne's testimony that convinces Sansa to trust Jaime and save his neck. "He is a man of honour," she swears before Sansa, Dany, and Jon, telling the Lady of Winterfell specifically that without Jaime, she would have never been able to protect Sansa when the time came. But it's the oath to Sansa's mother that Brienne brings up that takes us back to the seventh episode of season two, when the Stark family is still alive (except Ned, who dies back in season one) and Jaime swears an oath to Catelyn Stark.
At this point, Theon and the Ironborn are taking over Winterfell, Robb is in King in the North mode in the Westerlands, and Jaime is being held prisoner in Robb's camp. Robb contemplates what to do with his Lannister hostage: his bannermen want to kill him while Catelyn pleads with her son to let him go. Since they believe Sansa and Arya are both still in King's Landing (when in reality, Arya has long escaped), she wants to trade him in for her daughters. Robb refuses his mother's request, saying Sansa and Arya are "just girls" and thus not worth losing a valuable hostage over. (Yeah, Robb is a bit of a d*ck.)
When Robb travels to the Crag with Lady Talisa, Catelyn is left to deal with the bannermen's rage when Jaime kills one of their own in an attempt to escape. She delays their attempts to kill him and visits him during the night. The two have one of the most important conversations in the entire series as Catelyn berates Jaime for being a "man without honour," and in return, he explains the many ways that his "honourable" vows contradict one another. He also taunts the mother of five, pointing out how he's only been with Cersei and thus has more honour than Catelyn's own husband, who fathered a bastard.
Despite the tension between the two, Catelyn is there on a mission. She decides to free Jaime, but only if he swears to send Sansa and Arya back to her. She has him escorted by Brienne, who is unquestionably loyal to her, and Cleos Frey, who will help smooth the transaction for everyone. Jaime agrees, even though they both know this is really an act of desperation by a mother and Catelyn can't do anything if Jaime decides not to go through with it.
Of course, we sadly know how this story goes: Catelyn and Robb are murdered at the Red Wedding, Jaime returns to King's Landing only for his eldest son to be killed by poisoning, and Sansa escapes the South with the aid of Littlefinger. When an enraged Cersei demands that Jaime hunt down Sansa, he instead tasks Brienne with finding and protecting her. "We swore an oath to return the Stark girls to their mother . . . There's still a chance to find Sansa and get her somewhere safe," he tells her. He gives her his Valyrian sword forged from Ned Stark's sword Ice and offers Podrick as her squire.
In the end, neither Stark daughter returns to Winterfell with the aid of Jaime or Brienne — though Brienne does run into them both individually and offers her assistance. Sansa returns to her home as part of a betrothal with Ramsay Bolton, whom she later escapes from with the aid of Theon, and convinces Jon to go to battle against. While Brienne does pledge her sword to Sansa after she escapes Ramsay, it's safe to say that Sansa would have made her way home regardless. The same goes for Arya, who journeys to Winterfell on her own after becoming a Faceless Man.
So Jaime and Brienne didn't really keep their oath to Catelyn, because her badass daughters went and rescued themselves. But they did try! Considering how Jaime started out in the beginning of the series, that's definitely better than we could have hoped for.