What do you all think Yui was trying to teach Yuuya? I think the core hint for this is where she responds to Yuuya saying the unit is "junk" - when she brings up "jinba ittai" (horse and rider as one, or words to that effect) and hopes Yuuya would be able to appreciate that. It seems Yui is happy enough with Yuuya's skills but feels like he's about to hit a wall in his progress or that there's a factor that's slowing down his progress. Basically, though Yuuya wants to master the machine he also hates it and distrusts it - he's only seeing the negatives and not seeing the positives.
On a side note, the first time I remember hearing "jinba ittai" was when Mazda started using the phrase regarding the development of a new MX-5 / Miata
It's also interesting to consider - if Yui's so clearly capable a pilot, why couldn't she be the test pilot herself. I've posted before about why we might have a US pilot trying to learn how to pilot a new TSF designed for Japanese pilots - it's not very efficient is it. It hasn't been started in the anime so far but I'm still guessing it was at the request of the Americans involved - they may have gone so far as to refuse to co-operate unless the main test pilot was American. If the Americans' true motivation for project prominence was to learn more about foreign TSF design then they would probably be delighted when Yui offered to use her Takemikazuchi as part of a "special" training session...
On a side another note, the following thought occurred to me: it's not uncommon for the "hero" in a series to have some "grouchy old mentor" type figure to help him learn - someone the hero tends to hate or resent... Kinda seems they almost want to make Yui into such a mentor figure... then turn her into love interest
