Here's the picture I have in mind. In the case of presheaves of sets on a topological space $X$ there's an adjunction $$\Gamma\vdash \Lambda$$ between functor $\Lambda:\textbf{Psh}(X)\to \textbf{Top}/X$ associating to $P$ its étale bundle, and $\Gamma:\textbf{Top}/X\to \textbf{Psh}(X)$ taking a morphism $f:Y\to X$ to the sheaf of sections.
I would like to export this adjunction to the case of sheaves of abelian groups, replacing $\textbf{Top}/X$ with the category of surjective bundles $p:Y\to X$ with the property that any fiber $p^{-1}(x)$ has a structure of abelian group such that sum $+:Y\times_XY\to Y$ and inverse $()^{-1}:Y\to Y$ are continuous functions.
First of all, is it true the claim that there is an adjunction of these two functors seen as functors between the category of sheaves of abelian groups and the category of these bundles? It seems to be in Mac Lane - Moerdijk, but it's quite just mentioned.
Second, I'm trying to see at least that these functors are well defined, and on one hand it's easy to see that the sheaves of sections of such a bundle are actually sheaves of abelian groups, because we can defined pointwise sum of sections. However, I'm stuck in trying to prove that the étale bundle enjoy continuity of group operations. More precisely, a basic open $\mathcal{U}=\{(x,s_x)|x\in U\}\subseteq\Lambda(P)$, where $U\subseteq X$ and $s\in F(U)$ are fixed, has preimage under $+$ given by $\{(y,y')\subseteq \Lambda(P)\times_X\Lambda(P)|x:=f(y)=f(y')\in U\land y+y'=s_x\}$. Why is this set open in the topology of $\Lambda(P)\times_X\Lambda(P)$?
The case of the inverse function is actually clear, since $()^{-1}$ maps $\mathcal{U}$ to itself because germ of the inverse is the inverse of the germ, being cocone morphisms (to each stalk) group homomorphisms.
Thanks in advance for any suggestion!