Past exam of the mathematics course of the University of Cambridge / 2009 / iii / Paper 23 / 5 / c

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.