An example of a coproduct of sheaves in the category of presheaves that is not a sheaf

Question

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 $I$?

Answer 1

The empty coproduct is such an example if the site admits empty covers.

Concrete examples come from sheaves on topological spaces which must always send the empty set to the one point set.

In particular this means that only the "unary" coproduct of sheaves in $\text{PSh}$ is a sheaf.

Answer 2

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.