 Solution

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.