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Kate and Yelena start off as enemies, as Kate thinks Yelena wanted to kill her. As Barton was guarding the rooftop, Yelena attacks him. At that moment, Kate comes to his aid. And then is when Yelena throws Kate off the roof, which makes Kate think she wanted to kill her. Barton rushes to save her so she survives. Kate aims her bow at Yelena, but since she doesn't shoot, that allows Yelena to retreat. However, her fear of Yelena will not subside, even more so when she finds out Yelena is a Black Widow assassin.
So her terror since Yelena broke into her apartment the next time they meet is understandable. Kate throws a bottle at Yelena, that she gracefully catches and Yelena invites Kate to eat some macaroni she made herself. Kate angrily scoffs at this telling her she won't just sit to have lunch with the person who tried to kill her. After assuring Kate that she's not a threat, Yelena sits at the table with her, enjoying her meal, as Kate watches her cautiously. As she eats, she opens up to Kate and tells her that her sister was Natasha Romanoff, and that she saved the world. This shocks Kate, making her glad that she didn't shoot Yelena that day. After their meal, Yelena goes out of Kate's apartment through the window.
After this encounter, they meet each other again at Kate's mom's party. Previously, Yelena trusted Kate enough to tell her she was hired by Kate's mom to kill Barton. And now, she just tells Kate that she's there to "kill Barton, have some appetizers" and then leave. So after hearing this, Kate and Yelena have an intense fight where Kate wants to prevent Yelena from going to where Barton is, to kill him. Finally, Yelena jumps through the window, escaping once again.
Katelena is one of the most popular ships to come out of the Hawkeye series. Most people ship the two as best friends and would love to see them as a team. After fans noticed the great chemistry the two had in the series, they hoped for a series focused just on the two of them, as the new Hawkeye and Black Widow. On the other hand, some fans also ship them romantically. Although this take has more push back from other fans who point out the fact that Yelena is canonically aro/ace in the comics.