While working the puzzle above I ran into two questions:

1. When you connect the commuting diagrams between \$$\cup \leftrightarrow \cap^\ast \text{ and } \cap \leftrightarrow \cup^\ast\$$ using the \$$(-)^\ast\$$ functor, the diagram kind of makes it looks like \$$(-)^\ast\$$ is a self adjoint. Does such a thing exist?

2. In order to show that the diagram commutes, we had to prove two composite morphisms are equal. So for fun I tried expanding this out into a square as below:

![identity dual](http://aether.co.kr/images/identity_dual.svg)

Is the identity morphism an isomorphism? Can it have a dual that is not isomorphic to itself?