= Five adjoints for constant presheaves on open sets
{title2=$\Lambda\dashv\operatorname{ev}_{\varnothing}\dashv\Delta\dashv\operatorname{ev}_S\dashv\nabla$}
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>.
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