I am referring to this claim from Görtz and Wedhorn, page 49:
I get the intuition provided by the paragraph, but I absolutely do not get how this intuition is satisfied by the inverse limit. Intuitively I want to define $\mathscr{F}(V)$ using something involving intersections and disjoint unions. Perhaps I am missing an unstated isomorphism here, but how can it be the case that the section of $\mathscr{F}$ on $V$ is composed of tuples of sections on $U \in \mathcal{B}$? Excuse my inner programmer, but there seems to be a type mismatch here.
Also, what is the point of requiring $s_U \mid _{U'} = s_{U'}$? Is it just the "most universal" way to extract sections that agree on $V$?
In terms of your basis $\mathcal{B}$ you cannot express the gluing condition as $s_U|_{U\cap V}=s_V|_{U\cap V}$ since $U\cap V$ may not be a basis element whenever $U$ and $V$ are. But $U\cap V$ is a union of basis elements, so we can replace that by $s_U|_{W}=s_V|_{W}$ where $W$ runs through the basis elements $\subseteq U\cap V$. But a particular case of this is $s_U|_W=s_W|_W=s_W$. So the gluing condition is equivalent to $s_U|_W=s_W$ for all basis elements $U\supseteq W$.
We can regard $\mathcal{B}$ as a poset, and so a category and $U\mapsto \mathscr{F}(U)$ as a functor from $\mathcal B$ to $\mathbf{Ab}$. From the description of inverse limits of inverse systems of Abelian one get the authors' equality.