Pronouns can be so confusing. Is this appropriate or is there a preferred structure other than this?
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Your sentence is perfectly correct. Jamie and I is right because it's a subject (Jamie and me is a common mistake) and you and Kate is correct too. Both are in a natural order, as well: it's common to put I, me, or myself last in a list and you or yourself either first or last. I and Jamie is correct but odd, and Kate and you sounds pretty strange; many people write something like Kate and yourself as an alternative.