= Solution
Let $\widehat{\mathcal O}=[\mathcal O(S)^{\mathrm{op}},\mathbf{Set}]$, with <categorical presheaf> restrictions $X(U)\to X(V)$ when $V\subseteq U$. The five <functors> are
$$
\Pi X=X(\varnothing),\qquad \Gamma X=X(S),\qquad
(\Delta A)(U)=A,
$$
$$
(\Lambda A)(U)=
\begin{cases}A&U=\varnothing,\\\varnothing&U\ne\varnothing,\end{cases}
\qquad
(\nabla A)(U)=
\begin{cases}A&U=S,\\\{*\}&U\ne S.\end{cases}
$$
In $\Delta A$ all restriction maps are identities. In $\Lambda A$ and $\nabla A$, all nonidentity restrictions are the unique maps permitted by the displayed <sets>. These definitions are valid also when $A$ is empty: a restriction out of $(\nabla A)(S)$ still has singleton target. Maps of <sets> act at the displayed $A$-components, and <natural transformations> act on $\Pi$ and $\Gamma$ by evaluation.
A <natural transformation> $\Lambda A\to X$ is determined by an arbitrary map $A\to X(\varnothing)$, since all other source components are empty. A <natural transformation> $X\to\Delta A$ is determined by its empty-open component $X(\varnothing)\to A$: the component at $U$ must first restrict $X(U)\to X(\varnothing)$. Similarly, a <natural transformation> $\Delta A\to X$ is determined by $A\to X(S)$, followed by the restriction $X(S)\to X(U)$. Finally, a <natural transformation> $X\to\nabla A$ is determined by $X(S)\to A$, since all proper-open targets are singletons. These descriptions give the <hom-set> bijections
$$
\widehat{\mathcal O}(\Lambda A,X)\cong\mathbf{Set}(A,\Pi X),\qquad
\mathbf{Set}(\Pi X,A)\cong\widehat{\mathcal O}(X,\Delta A),
$$
$$
\widehat{\mathcal O}(\Delta A,X)\cong\mathbf{Set}(A,\Gamma X),\qquad
\mathbf{Set}(\Gamma X,A)\cong\widehat{\mathcal O}(X,\nabla A).
$$
Thus the <five adjoints for constant presheaves on open sets> are
$$
\boxed{\Lambda\dashv\Pi\dashv\Delta\dashv\Gamma\dashv\nabla.}
$$
The values at the empty open set are genuine <presheaf> values here; no <sheaf> condition or sheafification is being imposed.
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