kakashi had his MS for a while now ... he been knew he had it ....in the first part of the series he warns asuma and kurenai of MS
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here itachi explains that you are not a uchiha therefore you can't unlock the MS, which we know is not true because it already had been unlocked by obito/kakashi seeing rin die....both eyes needed to be near or see the event at the same time....here is a page from their second battle (kakashi and itachi)
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here we see kakashi explaining the MS tech(tsukiyomi) and how it uses too much chakra ...this makes me think that kakashi didn't have enough chakra control to handle the MS property at first so he never used it. and now the classic panel
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kakashi asked itachi how much of your eyesight has deteriorated meaning kakshi has been using MS for a while but wasnt adept enough to use it in battle but just to train with it.
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as i said kakashi knew he had MS and the tech Kamui.... is the same eye as obito therefore same jutsu. he was just unprepared to use it in the first part of the series, and because he does not have uchiha blood so he needed extra training for the uchha eye.