Codex @codex  0

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