= Solution
For a <categorical presheaf> $P$ on the open subsets, define the <functors> on objects by
$$
L(P)=P(\varnothing),\qquad R(P)=P(S).
$$
To prove $L\dashv\Delta$, a <natural transformation> $t:P\to\Delta A$ must satisfy
$$
t_U=t_{\varnothing}\circ\operatorname{res}^{U}_{\varnothing}
$$
for every open $U$, since the restrictions of $\Delta A$ are identities. Thus it is determined by the <function> $P(\varnothing)\to A$. Conversely such a <function> defines all $t_U$ by that equation, and the <categorical presheaf> restriction laws make the components compatible. Hence
$$
\operatorname{Nat}(P,\Delta A)\cong\operatorname{Set}(P(\varnothing),A).
$$
For the other <adjunction>, a transformation $s:\Delta A\to P$ is determined by its component $s_S:A\to P(S)$, since
$$
s_U=\operatorname{res}^{S}_U\circ s_S.
$$
Every <function> $A\to P(S)$ supplies compatible components by this formula. Therefore $\operatorname{Nat}(\Delta A,P)\cong\operatorname{Set}(A,P(S))$, giving the <adjoints to constant presheaves on open sets>:
$$
\boxed{\operatorname{ev}_{\varnothing}\dashv\Delta\dashv\operatorname{ev}_S.}
$$
These are adjoints of the stated constant-presheaf <functor>, with its value at every <open set> including the empty one; no sheafification is involved.
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