= 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:
$$
\left(\bigvee_i f_i\right)\left(\bigvee_j a_j\right)
=\bigvee_{i,j}f_i(a_j).
$$
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 gives
$$
[B^*,A^*]\cong[A,B].
$$
It is order-preserving because taking a right adjoint reverses pointwise order once, while the order on $A^*$ reverses it again.
A join map $\chi_a:A\to2$ is associated with $a\in A$ by
$$
\chi_a(x)=0\Longleftrightarrow x\leq a.
$$
Every join map to $2$ is of this form, and $a\mapsto\chi_a$ identifies $A^*$ 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 naturally
$$
\boxed{[A,[B,C]]\cong[B,[A,C]].}
$$
The assumed expression of $C$ as a limit of copies of $2$ reduces the verification to the preceding $2$-valued description.
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