In Etale cohomology theory, by Lei Fu, Page, 195, it is stated that given a strictly Henselian ring $A$ with closed point $s$, then $\mathrm{Spec}A$ is a final object in the opposite category of étale neighborhoods of $s$, so that for any étale sheaf (say, over $X$) $\mathscr{F}$, $\mathscr{F}_s$ is $\mathscr{F}(\mathrm{Spec}A)$.
The given proof is a reference to the following two facts:
- For an unramified separated morphism $f: X \to Y$ with $Y$ connected, a section of $f$ is uniquely determined by its value at a single point
- That for an étale morphism $f: X \to \mathrm{Spec}A$, any section $f_s: s \to X \times_A s$ is deduced by base change from a section $\mathrm{Spec} A \to X$ (in this fact, A simply has to be Henselian and not strictly Henselian).
There are several things I don't understand here: the first being, in what way is $\mathrm{Spec}A$ an étale $X$-scheme? Can we find an étale morphism $\mathrm{Spec}A \to X$ over $s$ because $A$ is strictly henselian? Is such morphism directly constructed or is it taken as a section of a morphism $X \to \mathrm{Spec}A$ (in this case, how to construct it?)
And finally, how does this imply that $\mathrm{Spec} A$ is final in the opposite category of étale neighborhoods of s, i.e that for any étale neighborhood $U$ over $s$, then one can find a morphism $\mathrm{Spec}A \to U$?
I suppose the construction I ask for in the first question would give an answer to that second question, as it would be enough to repeat it for $U$ instead of $X$, the question being etale local near $s$.
Edit:
By adapting an argument a few lines below, I have managed to reduce the case to show that it indeed suffice to show that $\mathrm{Spec}A$ is an étale $X$-scheme:
One has that $$\begin{align} \mathrm{Hom}_X(\mathrm{Spec}A, U) &\cong \mathrm{Hom}_{\mathrm{Spec}A}(\mathrm{Spec}A, \mathrm{Spec}A \times_X U)\\ \end{align} $$ The latter is canonically isomorphic to $\mathrm{Hom}_s(s, s \times_{\mathrm{Spec}A} \mathrm{Spec}A \times_X U)$ because $A$ is strictly henselian: because $U$ and $\mathrm{Spec}A$ are étale $X$-schemes,the morphisms in $\mathrm{Hom}_{\mathrm{Spec}A}(\mathrm{Spec}A, \mathrm{Spec}A \times_X U)$ are etales, and since $A$ Henselian, the sections are surjective on the set of s-morphisms $s \to s \times_{\mathrm{Spec}A}\mathrm{Spec}A \times_X U$, and since $A$ is strictly Henselian, and we can assume the morphism is also separated, by replacing U with a smaller neighborhood (affine for example), since we ultimately want to show that $\mathrm{Spec}A$ is final, it doesn't impact the proof (however Fu acts as if separation wasn't an issue to show that the map $\mathrm{Hom}_{\mathrm{Spec}A}(\mathrm{Spec}A, U) \to \mathrm{Hom}_s(s, s \times_{\mathrm{Spec}A} U)$ is injective, why that?).
As $\mathrm{Hom}_s(s, s \times_{\mathrm{Spec}A}\mathrm{Spec}A \times_X U)$ canonically isomorphic to $\mathrm{Hom}_s(s,s \times_X U)$, which is canonically isomorphic to $\mathrm{Hom}_X(s, U)$, we are done.
So it really boils down to showing that $\mathrm{Spec}A$ is an etale $X$-scheme. I haven't made any progress in that direction. Also, is the above correct?