The constant presheaf need not be a sheaf on a disconnected open set

By [F1], ${\underset{‾}{A}}_{\mathrm{p}\mathrm{r}\mathrm{e}}$ is a presheaf. On the disjoint cover $U=U_1\cup U_2$, choose the local sections $s_1:=a\in {\underset{‾}{A}}_{\mathrm{p}\mathrm{r}\mathrm{e}}(U_1)$ and $s_2:=b\in {\underset{‾}{A}}_{\mathrm{p}\mathrm{r}\mathrm{e}}(U_2)$. Both restrict to the unique element of ${\underset{‾}{A}}_{\mathrm{p}\mathrm{r}\mathrm{e}}(\mathrm{\varnothing })$, so the pair is compatible on $U_1\cap U_2=\mathrm{\varnothing }$.

A glued section over $U$ would have to be an element $s\in A$ whose restrictions to the two nonempty opens are $a$ and $b$. Those restriction maps are identities, so this would force $s=a=b$, contradicting the choice $a\mathrm{\ne }b$. Therefore ${\underset{‾}{A}}_{\mathrm{p}\mathrm{r}\mathrm{e}}$ is not a sheaf.

Write $P={\underset{‾}{A}}_{\mathrm{p}\mathrm{r}\mathrm{e}}$. At any point $x$, two constant sections on neighbourhoods of $x$ have equal germs exactly when their values in $A$ agree: restriction to a smaller neighbourhood is the identity. Thus a germ-compatible presentation for $P^{+}(V)$ assigns a constant value on each nonempty chart, with equal values wherever two charts meet. These values define a locally constant function on $V$. Equivalent presentations define the same function, and presentations defining the same function are equivalent pointwise. Every locally constant function has such a presentation by constant neighbourhoods. For $V=\mathrm{\varnothing }$, all presentations are equivalent and correspond to the unique empty function. These bijections commute with restriction, so $P^{+}\cong {\underset{‾}{A}}_{\mathrm{l}\mathrm{o}\mathrm{c}}$.

For any sheaf $G$, germ-compatible local sections agree on each overlap: equality of germs gives equality on a neighbourhood of each point, and sheaf locality gives equality on the whole overlap. They therefore glue uniquely to a section of $G(V)$. This gluing identifies $G^{+}(V)$ with $G(V)$, including the empty open, naturally in $V$. Apply this to the sheaf $G={\underset{‾}{A}}_{\mathrm{l}\mathrm{o}\mathrm{c}}$ from [L1]. Step 3.1 then gives $aP=P^{++}\cong ({\underset{‾}{A}}_{\mathrm{l}\mathrm{o}\mathrm{c}}{)}^{+}\cong {\underset{‾}{A}}_{\mathrm{l}\mathrm{o}\mathrm{c}}$; the single-chart unit sends a constant section to its constant function.