TITLE: An example of a coproduct of sheaves in the category of presheaves that is not a sheaf
QUESTION [5 upvotes]: For a site $C$ there is an adjunction $a\colon PSh(C)\leftrightarrows Sh(C)\colon \iota$ with sheafification $a$ and inclusion $i$. $a$ preserves finite limits. The inclusion $\iota$ does not preserve finite colimits in general.
Let $\{F_i\}_{i\in I}$ be a family of sheaves. There coproduct $\coprod_{i\in I} F_i$ in $Sh(C)$ is given by $a\left( \coprod_{i\in I} \iota(F_i) \right)$, i.e. the sheafified presheaf-coproduct of the $F_i$s.

What is an example of such a presheaf-product $\coprod_{i\in I} \iota(F_i)$ that is not already a sheaf if all the $F_i$s were sheaves? Is this always true for a finite set $I$?

REPLY [3 votes]: For topological spaces (and probably something similar will hold for many sites) you can explicitly write down the coproduct of sheaves (which I assume to be $\mathsf{Set}$-valued here):
$$\left(\coprod_i F_i\right)(U) = \left\{\bigl((U_i)_i,(s_i)_i\bigr) : U = \coprod_i U_i,\,s_i \in F_i(U_i)\right\}$$
Here you can see that there is a huge difference between the coproduct of sheaves and the coproduct of the underlying presheaves! This is even more clear if you work with representable sheaves:
$$\coprod_i \hom(-,X_i) = \hom(-,\coprod_i X_i)$$
in the category of sheaves, but of course this is far from being true "pointwise", i.e. for the underlying presheaves.