Is it correct to say, "Jamie and I look forward to seeing you and Kate on Friday?"
Solution 1:
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.