Corollary 5.3.4 in Szamuely, Galois group and fundamental groups

Question

There is a specific point in the proof of Corollary 5.3.4 in Szamuely, Galois group and fundamental groups that I can't seem to fully understand .

Specifically the statement is

Corollary 5.3.4.

If $:⟶$ is a connected finite etale cover, the nontrivial elements of $Aut(|)$ act without fixed points on each geometric fibre. Hence $Aut(|)$ is finite .

The proof of this corollary seems to be an immediate consequence of the following :

Corollary 5.3.3.

If $⟶$ is a connected -scheme and $_1,_2:⟶$ are two -morphisms to a finite étale -scheme with $_1∘=_2∘$ for some geometric point $:Spec(Ω)⟶$, then $_1=_2$.

From this statement it is clear that any geometric point of $X$ can't be fixed by any element of $Aut(|)$ but I still wonder how this implies that the action of $Aut(|)$ on $X×_SSpec(Ω)$ can't fix any points of the fibre where $:Spec(Ω)⟶S$ a geometric point .

There was another mathstackexchange question posted regarding that proof:

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but it doesn't seem to answer my question .

Thank you very much for any help!