Suppose $A$ is a complete Heyting algebra and $a : \widehat{A} \to \text{Sh}(A)$ is the associated sheaf functor (where the topology on A is the usual sup topology).
Is there a simpler description of $a$ than applying the $^+$ construction twice? If not in general, suppose in addition the order on $A$ is total. Does that help?
For example, if $A$ is a partial order and $\text{Idl}(A)$ is the complete Heyting algebra of downwards closed sets of $A$, then for $X \in \widehat{\text{Idl}(A)}$ and $b\in A$, we have $a(X)(\downarrow b) = X(\downarrow b)$, where $\downarrow b = \{ x \in A\ {\large|}\ x \leq b \}$, but these are in general not all the downwards closed sets, of course.
