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For a presheaf of sets on a topological space $P$, define $LP=P(\varnothing )$ and $RP=P(S)$. Then the adjoints to constant presheaves on open sets areTo prove the first adjunction, a natural transformation $a:P\to \Delta A$ must satisfy $a_U=a_{\varnothing }\circ {\mathrm{res}}_{U,\varnothing }$ by naturality at the inclusion $\varnothing \subseteq U$. Hence it is uniquely determined by a function $P(\varnothing )\to A$. Every such function defines a natural transformation by this formula, because restriction maps compose. This gives $\mathrm{Nat}(P,\Delta A)\cong \mathbf{Set}(P(\varnothing ),A)$.

For the second adjunction, a natural transformation $b:\Delta A\to P$ satisfies $b_U={\mathrm{res}}_{S,U}\circ b_S$, so it is uniquely determined by a function $A\to P(S)$. Every such function defines the remaining components by restriction, giving $\mathrm{Nat}(\Delta A,P)\cong \mathbf{Set}(A,P(S))$. Evaluation acts on presheaf morphisms componentwise, so these object assignments are indeed functors. The use of $P(\varnothing )$ is essential: a constant presheaf of sets has value $A$ there, whereas a sheaf has a singleton value there. No sheaf condition is imposed in this question.

For a categorical presheaf $P$ on the open subsets, define the functors on objects byTo prove $L⊣\Delta$, a natural transformation $t:P\to \Delta A$ must satisfyfor 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. HenceFor the other adjunction, a transformation $s:\Delta A\to P$ is determined by its component $s_S:A\to P(S)$, sinceEvery function $A\to P(S)$ supplies compatible components by this formula. Therefore $\mathrm{Nat}(\Delta A,P)\cong \mathrm{Set}(A,P(S))$, giving the adjoints to constant presheaves on open sets:These are adjoints of the stated constant-presheaf functor, with its value at every open set including the empty one; no sheafification is involved.