Past exam of the mathematics course of the University of Cambridge / 2003 / iii / Paper 18 / 6 / i / Solution

A presheaf on $0\to 1$ is an arrow $r:B=X(1)\to A=X(0)$ in sets. A morphism of presheaves is a commuting square. DefineA commuting square from $r$ to $\Delta S$ is determined by its map $A\to S$, while a square from $\Delta S$ to $r$ is determined by its map $S\to B$. A square from $r$ to $\nabla S$ is also determined by its map $B\to S$, because the other component is the unique map $A\to 1$. Therefore these Hom-set bijections establish the adjoint chain for evaluation in an arrow category:There is a further left adjoint to $\Pi$: $\Lambda (S)=(\varnothing \to S)$. A square $\Lambda S\to r$ is exactly a map $S\to A$, proving $\Lambda ⊣\Pi$.

There is no right adjoint to $\nabla$. If there were, $\nabla$ would be a left adjoint and preserve the initial object. But it sends the empty set to $(\varnothing \to 1)$, whereas the initial object of this arrow category is $(\varnothing \to \varnothing )$. They are not isomorphic. Hence $\Pi$ does have a left adjoint, while $\nabla$ has no right adjoint.