= Solution
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 adjoint
$$
f^*(b)=\bigvee\{a:f(a)\leq b\}.
$$
A right adjoint preserves arbitrary meets. Regard it as the join-preserving map
$$
f^*:B^{\mathrm{op}}\longrightarrow A^{\mathrm{op}}.
$$
Thus define $A^*=A^{\mathrm{op}}$ and send $f$ to its right adjoint. Since a left adjoint is the right adjoint of its right adjoint after reversing orders, $(f^*)^*=f$ and $(A^*)^*=A$. This gives the involutive self-duality
$$
(-)^*:\mathbf{CSLat}^{\mathrm{op}}\longrightarrow\mathbf{CSLat}.
$$
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