We know that any Galois category is equivalent to the category of finite $\pi$-sets where $\pi$ is a unique profinite group. I think axioms for a Galois category were choosen so that the above theorem becomes true. But how would someone guess exactly axioms will give us the correct answer? Is there some kind of motivation behind choosing each axiom?
SGA I, Le groupe fondamental : généralités. Exposé V.