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You can get sick from nearly anything. Some people could get sick at “the idea of a dead fly in the bottom of a coffee cup”, and some could get sick from seeing it. Some could eat the fly deliberately and have no problem at all. (And some people will get a queasy feeling from just reading that.)
Assuming the water was clean to begin with and no other pathogens or toxicity occurred, I doubt very seriously that you’d have a problem from the fly. What I mean by that is that if the fly drowned or was killed by some other (nontoxic) means and deposited in the cup, then there’s probably no problem. But if something in the coffee killed the fly, then it’s the coffee itself that could hurt you, too.