Limit of finite sets

I've recently come across the following curious little result. I know how to prove it and have a use for it, but my question is: can anyone supply a wider context or explanation? The result is that the limit in Set of any diagram ... ---> S_3 ---> S_2 ---> S_1 of finite nonempty sets is nonempty. Note that finiteness cannot be dropped: for instance, take each S_n to be the natural numbers and each map to be addition of 1. Thanks, Tom