We see that he was able to catch the fly with chopsticks with very little practice and during the tournament even Miyagi was surprised by Daniel results:
Daniel: "I never thought I'd get this far."
Miyagi "Make two of us."
I suppose we should assume that Daniel was particularly suited and even a bit naturally gifted when it came to Karate. From the actor's standpoint though, I was stunned at how fast Ralph Macchio was able to move his feet when he was simulating Muhammad Ali's shuffle.
The fly and chopstick thing though is pure luck and was put in for humor's sake.
I think Miyagi could have been being funny with the "make two of us" quip. He believed in Daniel, at least enough to advance far enough to win the Cobra's respect.
I think he would have to be to learn that quickly.
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