Is there an example of a self-dual complete category that is not a partially-ordered set?



Self-dual Complete Category
The category of suplattices (complete join-semilattices) is complete and self-dual. It is complete because it is monadic over Set (the monad is the covariant powerset monad). The duality $Sup^{\mathrm{op}}\to Sup$ sends a suplattice $X$ to the opposite poset $X^{\mathrm{op}}$, which is again a suplattice since every suplattice is also an inflattice (define $\bigwedge S = \bigvee \{ x | x \le S\}$). It sends a sup-preserving map $f:X\to Y$ to $(f^\ast)^{\mathrm{op}}:Y^{\mathrm{op}}\to X^{\mathrm{op}}$, where $f^\ast:Y\to X$ is the right adjoint of $f$—this exists by the adjoint functor theorem for posets, and preserves infs since it is a right adjoint.


