Is Set, the category of sets, a self-dual category?

Question

Because Rel, the category of relations, is self-dual, and because Rel has Set, the category of sets, as a subcategory, can we conclude that Set itself is self-dual? And, just as Rel is a dagger category, is Set also a dagger category?

Answer 1

No, $\text{Set}$ and $\text{Set}^{op}$ differ in many ways. One of the simpler ones is that $\text{Set}$ is a distributive category: finite products distribute over finite coproducts. This is very false in $\text{Set}^{op}$, as can already be seen for finite sets. Another is that $\text{Set}$ is generated by its terminal object; this is again very false in $\text{Set}^{op}$.