Adjoints to constant presheaves on open sets (${\mathrm{ev}}_{\varnothing }⊣\Delta ⊣{\mathrm{ev}}_S$)

For a topological space $S$, let $\Delta A$ be the constant presheaf of sets on its open-set category, with identity restriction maps, including the value at the empty open set. A natural transformation $P\to \Delta A$ is determined by its map $P(\varnothing )\to A$; its component at $U$ composes with $P(U)\to P(\varnothing )$. A transformation $\Delta A\to P$ is determined by $A\to P(S)$; its component at $U$ restricts sections from $S$. These two bijections prove the displayed adjunctions. This concerns presheaves, so no sheafification is involved.

For a nonempty topological space $S$, the constant diagram functor on its open-set presheaf category lies in this five-term adjunction chain. The leftmost functor sends $A$ to the categorical presheaf equal to $A$ at the empty open set and empty elsewhere. The rightmost sends $A$ to the presheaf equal to $A$ at $S$ and a singleton elsewhere. A natural transformation from the former is determined by its empty-open component; a transformation into the latter is determined by its whole-space component. The middle adjunctions follow from restriction maps to the empty open set and from the whole space. Neither endpoint extends: the leftmost functor fails to preserve the terminal object, and the rightmost fails to preserve the initial object.