Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 119 / 3

The adjoint functor theorem for complete lattices says that a monotone map $f:A\to B$ between complete lattices preserves arbitrary joins exactly when it has a right adjointA right adjoint preserves arbitrary meets. Regard it as the join-preserving mapThus define $A^{\ast }=A^{\mathrm{op}}$ and send $f$ to its right adjoint. Since a left adjoint is the right adjoint of its right adjoint after reversing orders, $(f^{\ast }{)}^{\ast }=f$ and $(A^{\ast }{)}^{\ast }=A$. This gives the involutive self-duality

Let $[A,B]$ be the set of arbitrary-join-preserving maps $A\to B$, ordered pointwise. Pointwise joins remain join-preserving because the two joins may be interchanged:Precomposition and postcomposition preserve these joins, so $(A,B)\mapsto [A,B]$ is the required $\mathbf{CSLat}$-valued hom functor.

Sending $f:A\to B$ to its right adjoint givesIt is order-preserving because taking a right adjoint reverses pointwise order once, while the order on $A^{\ast }$ reverses it again.

A join map ${\chi }_a:A\to 2$ is associated with $a\in A$ byEvery join map to $2$ is of this form, and $a\mapsto {\chi }_a$ identifies $A^{\ast }$ with $[A,2]$. Finally, a map $A\to [B,C]$ is a function $A\times B\to C$ preserving arbitrary joins separately in each variable. Swapping the variables gives naturallyThe assumed expression of $C$ as a limit of copies of $2$ reduces the verification to the preceding $2$-valued description.