In Moerdijk, Classifying spaces and classifying topoi, on page 22, we have a functor from the topos $\mathcal BG$ of right $G$-sets ($G$ is a group) to the topos $Sh(X)$, the sheaves (étale spaces) over a topological space $X$. We want to prove that it preserves colimits and finite limits. The Author says that "it suffices to check this for the stalk at each point $x\in X$". He then computes the stalk $F(S)_x$ for every $S\in \mathcal BG$ (call $F$ the functor) and finds that $F(S)_x\cong S$ for every $x$, so it is clear that $F$ preserves colimits and finite limits.
My question is: why does it suffice to study stalks? My idea is that, as always, the sheaf structure allows one to "recover" the global properties later, and focus on what happens on stalks. Can one help me formalizing this in relationship to the specific context of limits and colimits?
Thank you in advance.