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Why did Jen show Doc Brown the note that was in her pocket? She knows what happened, plus she didn't even know she would see Doc there, so does she just carry it around in case she ever bumps into him again?
Plus, doesn't she know that it already changed at the end of the second movie?
The "You're Fired" memo from the future did not change at the end of Part II.
Jennifer wasn't even around for the end of Part II, which was entirely in 1955. Jennifer was still sleeping on her front porch after Doc and Marty dropped her there in the "alternate" 1985. She still had the memo on her because there wasn't any time for her to do anything else - Marty comes by to make sure she's OK once time goes back to normal, and she's just waking up.
The memo changed as part of the same sequence where Doc shows up, which is the end of Part III. Marty ditches the "chicken" race, and Jennifer remembers what she saw from the future, where Marty was injured in this same race. She pulls out the memo and it fades in front of her, and she confronts Marty with it, asking what happened and what is going on. He takes her to see the wreckage of the time machine to explain. That's where Doc makes his entrance. It's all literally a couple minutes at the end of the movie, not separated across two movies like you described.
The events, from Jennifer's perspective at any rate, are rather clearer in the film's official novelisation.
Due to the change in the timeline (caused by Marty not having an accident involving the Rolls Royce) Jennifer only now half-remembers seeing the fax come through. When she finds the paper in her pocket, the message on it disappears entirely just at the same time as her memories of the previous night, which now no longer happened, start to fade away. This explains her confusion.
This was more than just a dream. There was a piece of paper in her pocket, a paper she remembered.
She pulled out the crumpled page and looked at the huge letters at the centre:
YOU’RE FIRED.
There! It wasn’t a dream after all! It was—
The words disappeared from the page.
‘It erased!’ she exclaimed.
‘What?’ Marty asked, looking at the page over her shoulder. ‘What erased?’
Jennifer frowned. Why was she staring at a blank piece of paper? And why had that paper been in her pocket?
‘I don’t know,’ she admitted. What importance could a blank piece of paper have, anyway? She was with Marty, now, and they were going to have a wonderful time together. That was what was really important, after all.
When she encounters the Doc, with a head full of questions, a memory made out of swiss-cheese and absolutely no answers, she shows him the disappearing message note hoping that he'll explain to her why it's so important. His answer is not especially forthcoming.
‘Then it wasn’t a dream,’ Jennifer muttered at his side. He looked over to see her staring intently at a crumpled piece of paper that didn’t seem to have anything on it. She looked up at the inventor.
‘Doctor Brown, I just want to know one thing What happens to Marty and me in the future?’
Doc considered her question for a second before answering. ‘In the future? That’s up to you. Your future hasn’t been written - no one's has. For better or worse, your future is what you make it.’ He winked at both Marty and Jennifer ‘So make it a good one. Both of you.'
