> (1) chyotto matte tte
> (2) why does the tte mean "... and I'll be back shortly".
(1)
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I'll be back shortly
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(2) Why don't you wait for me a moment in spite of my saying "Please wait a moment"? I'm thinking " **I'll be back shortly.** "
If "(1) chyotto matte **tte** " is "(1)' chyotto matte **te** **** ", it is a short form of " **** ", which implies that " _Would you wait a moment, because I'll be back shortly?_ "
The second in **** is "an auxiliary indicates continuing action" according to Jisho.org here, so the phrase means "Keep waiting for a moment."
> chyotto matte X, where X could mean "I'll give food to my cat" or "I'll need sign an important document for Toyota" or "I'll fix my laptop which was broken by little sister"?
chyotto matte X
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"Wait a moment, please, because I'll do X".