In Butz, Moerdijk, Representing topoi by topological groupoids, it is proven that given a topos $\mathcal E$ with enough points there exists a topological groupoid $G\rightrightarrows X$ such that $\mathcal E=Sh_G(X)$.
The point I do not understand is where they say that an algebra structure $\tau$ on sheaf $F$ of $Sh(X)$ corresponds to an action of $G$ over $F$. It seems to me that, in that proof, the action is never required (or, in the other sense, proven) to be continuous, as the definition of $Sh_G(X)$ would require. In particular, if I have a continuous action, where does this appear in the corresponding algebra structure?
Thank you in advance.
EDIT: Sorry, I think the solution is pretty straightforward: the algebra structure is a morphism in $Sh(X)$, hence continuous. Conversely, if the action is continuous then what we would like to be an algebra structure is continuous, therefore is a morphism of sheaves.