I am trying to learn a bit about étale cohomology from Milne’s book and I have run into some early problems. I learned ordinary abelian sheaf cohomology from Hartshorne. There, sheaves are defined on the category of open subsets of a scheme $X$ in which the morphisms are just the open inclusions. A presheaf on $X$ is then just a contravariant functor on this category. To motivate the étale site and étale cohomology, many sources first introduce sites via the Zariski open immersions, but I am already confused by this.
The Zariski site on $X$ is defined to have objects $f \colon U \rightarrow X$ which are open immersions and morphisms given by maps $V \rightarrow U$ which commute with the structure morphisms of $U$ and $V$. Then a presheaf $\mathscr{I}$ is a contravariant functor on this category. With this definition, many sources, including Milne, call the group $\mathscr{I}(U)$ the sections over $U$. But already I don’t know what this means. In the case of Hartshorne this is fine. But in the case of the Zariski site, there could be many distinct open immersions from $U$ to $X$. So what then does $\mathscr{I}(U)$ actually refer to? Is there any reason to believe that $\mathscr{I}(f \colon U \rightarrow X)$ is equal or even isomorphic to $\mathscr{I}(g \colon U \rightarrow X)$ if $f$ and $g$ are distinct open immersions? I suspect in the sheaf case one can use the sheaf axioms to argue that they are isomorphic, but what about in the presheaf case?
The problem seems to be even worse when it comes to the étale site. In the case of the Zariski site, at least open immersions factor into an isomorphism and an open inclusion. But two étale morphisms $f \colon U \rightarrow X$ and $g \colon U \rightarrow X$ need not be related in any way as far as I can see. So what does $\mathscr{I}(U)$ mean, and is there any reason to think it is independent of the structure morphism?
I realise this may be a completely silly question, but I can’t see an obvious answer after a while thinking about it.
This is an abuse of notation. $F(U)$ is not just a function of $U$ as an object, it's a function of the morphism $f : U \to X$, which is being dropped from the notation. That is, it's shorthand for $F(U \xrightarrow{f} X)$.