J's father

K didn't go back by his own action. He did not time-jump. He disappeared because Boris went back and killed him. Then J went back to before Boris killed K.

Why J was still an agent is not clear. it can't be because he time-jumped and was a part of the third stream time traveling. That hasn't happened yet.

Maybe it's because K told him about there being some secrets of the Universe, and J had encountered Boris and was a part of the setup of Boris's time travel. J was right in the middle of the K-Boris conflict.

K from 1969 did not know J, But one of the Boris's did. And as K disappeared he may have been thinking about J. So maybe that all worked to retain J as an agent, and with a memory of K.

That's the best I can do for now.
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The story is king.

We are told there is always a death in that final scene. If not K, then it is his father. So my take is that the original timeline and the updated timeline both has his fathers death. In other words, to save his father his partner would have to die. J is the hero and chooses the altruistic way out... or rather, he accepts it.

The question is then; how was K a part of J's life up until the first MIB movie? My best answer is that he took care of him from afar. In the original timeline J complained that he grew up with no father.... so in the end of the day, the only thing that changed was that K now remembered meeting future J. Everything else is and unfolded as it had done before.

And this is not a trivial difference. The reason K was so damn unemotional was because he knew and kept this awful truth. A truth he no longer had to keep. Their adventure this time not only freed our world it also freed K.

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** I am normally not a praying man, but if you are up there, please save me Superman **