Following proof from Milne's Étale Cohomology (page 94) contains an equality I not understand.
Setting: assume $X$ is a variety (=absolutely reduced, irreducible scheme of finite type over base field $k$) which by assumtion can be embedded via $ j: X \hookrightarrow \overline{X} $ as an open subvariety of a complete/ proper variety $\overline{X}$ and $F$ sheaf on $X_{\text{et}}$.
Denote by $i: \overline{X}-X \to \overline{X}$ the canonical closed immersion. The proof of Proposition 1.29(a) poses the equality
$$ \Gamma(\overline{X} - X,i^*j_*F)) = \varinjlim \Gamma(V\times_{\overline{X}} X,F) $$
where the inductive limit runs over all etale $V \to \overline{X}$ containing $\overline{X} - X$ in their images.
Question: Why this equality holds? Note that the right term equals directly by definition to the global sections $ \Gamma(\overline{X} - X,i^{\#}j_*F))$ where $i^{\#}G$ is the inverse image presheaf exactly defined that way, but which in most cases is not a sheaf, whose associated sheaf/sheafification is $i^{*}G$.
Metaquestion: This raises in light of the initial question above, why the equality above holds, the more general question when the presheaf $i^{\#}G$ is already a sheaf in this setting?
More generally if $f: X \to Y$ is a morphism of schemes are there rather "natural" (ie not too exotic) assumptions on $f$ such that for any sheaf $G$ on $Y$ (im what topos ever we suppose to work; but say Zariki or etale for sake of simplicity) the inverse image presheaf $f^{\#}G$ is already a sheaf and therefore coincides with it's sheafification $f^*G$?
The most obvious case is when $f$ is an open immersion. More generally, but only for Zariski topos if $f$ is open, if I'm not confusing something. (The argument in that case should be that we can check sheaf property for open images of open set on the target (which would be cofinal wrt considered direct system) without passing to limit which tends to destroy sheaf property.
But what about (closed) immersions (as in the problem above)? Seems not to be clear why that should be the case. Do there exist some sufficient criteria?
At all, the metaquestion looks quite "natural", but unfortunately I nowhere found a rigorous discussion on that problem.
Or, to come back to original question, is there an other reason involved why above the equality should hold?
