Inverse limit defining etale fundamental group.

Question

Let $(S,s)$ be a connected scheme with geometric point $s$. In many places, I can find the etale fundamental group being defined as $$\varprojlim_{X \to S} \text{Aut}_S(X)$$ Where $X \to S$ ranges over the Galois coverings. How is this inverse limit defined? Given $Z \to Y$ (over $S$), I don't see why an automorphism of $Z$ over $S$ has to induce an automorphism of $Y$ over $S$.