I *still* (even after rereading it this weekend) do not feel like I've got a handle on what went on there. Which is the point of having Lucky be the narrator, but... so far (unless I really really overlooked something - which is possible), I don't feel like the reason she left came from the conversation they had at the end of #10. (I'm also not sure who it was he was planning to write a letter to, or exactly why Kate referred to herself as Hawkeye right after kissing Kazi (did she mean for him to hear her? Or was that mumbled to herself? If the former, why? If the later, does that mean she caught on that he might be a bad guy? Or just that she was pleased with herself for taking the chance to kiss him?)

kadollan thinks that he ended up killing Gil because he was there to kill Clint but saw Kate go to Clint's apartment and recognized her from earlier. Which... idk. Entirely possible, but I'm not sure the timeline works so well, since Kate had already left again and Clint had had time to go talk to Gil up on the roof.

Out of curiosity (enquiring minds want to KNOW! ;) ) - do you think that it was the conversation where she told him she wasn't the person he got to bitch at that made her decide to leave? Or a later conversation we only see the Lucky-version of?