When $X$ is a topological space, any sheaf $S$ on $X$ is a connected object of the category of presheaves $\widehat{\mathcal{O}(X)}$ in the sense that
$\mathrm{Hom}_{\widehat{\mathcal{O}(X)}}(S,A+B) \cong \mathrm{Hom}_{\widehat{\mathcal{O}(X)}}(S,A)+\mathrm{Hom}_{\widehat{\mathcal{O}(X)}}(S,B)$,
for all $A,B \in \widehat{\mathcal{O}(X)}$.
Every natural transformation $\phi : S \to A+B$ must land entirely in $A$ or in $B$ according to whether $\phi_\emptyset: 1 \to A(\emptyset) + B(\emptyset)$ lands in $A(\emptyset)$ or $B(\emptyset)$.
The same situation holds for sheaves on a locale, because these still evaluate to the one element set at the bottom element of the corresponding frame.
Is there a characterization for the broader class of sites such that their sheaves are connected presheaves?