I have a question on a detail from Huber's Etale Cohomology of Rigid Analytic Varieties and Adic Spaces regarding the equivalence between overconvergent sheaves and sheaves on the partially proper site. In the section 8.1, The Partially Proper Topology, in Chapter 8 Partially proper sites of rigid analytic varieties and adic spaces, I am confused about the proof of Proposition 8.1.4i.a).
Specifically, I am confused about the following point. Let $x$ be a maximal point of a taut, analytic adic space $X$. Let $F$ be a sheaf of abelian groups on $X$. Denote by $S$ the set of specializations of $x$. It seems that Huber computes the global sections of the restricted sheaf $F|S$ using the formula for the presheaf restriction, meaning, it does not seem we need to sheafify to compute this restriction; at least, otherwise, I do not see how he applies Lemma 8.1.5 to conclude (in the last couple lines of the proof), for instance, that we can identify
$$(v_*F)_x\cong \Gamma(S,F|S)$$
where $v$ denotes the identity map $X\rightarrow X_{pp}$ from $X$ to the associated space with the partially proper topology. Note that I don't think this part depends on $F$ being overconvergent, which rather comes in to identify the global sections $\Gamma(S,F|S)$ with the stalk $F_x$.
I can see that Lemma 8.1.5 tells us that the partially proper opens of $X$ containing $x$ form a fundamental system of open neighborhoods of $S$, but this would seem to help identify the stalk of $v_*F$ at $x$ with the global sections of the restriction as presheaves of $F$ to $S$, versus the restriction of sheaves.
Any clarifications on this point are much appreciated. I guess I wonder if the presheaf restriction of $F$ to $S$ is already a sheaf and thus agrees with $F|S$, or something else is going on. I have thought about the fact that $S$ is closed under generalization, but that alone did not seem to help.