adjunctions between FinCat and Set I am looking for an ambidextrous adjunction between FinCat and Set.  FinCat, I am defining as the category of all finite categories, with functors as morphisms.  Does such an adjunction exist?  I am most interested in the Frobenius Monad on Set that is generated by this adjunction.  My guess is that there should be "enough" limits and colimits in FinCat for this adjunction.
 A: No. Suppose we have a functor $F\colon \mathsf{(Fin)Set}\to \mathsf{(Fin)Cat}$ which is both a left and a right adjoint (the finiteness in both places is completely optional). Since $F$ is a right adjoint, it preserves limits, and in particular it preserves the terminal object. So $F(1)$ is the category with a single object (and a single arrow, the identity on that object). But since $F$ is a left adjoint, it preserves colimits. And every set is isomorphic to a coproduct of copies of $1$. So $F$ is uniquely determined up to isomorphism: $F(X)$ is a discrete category with $|X|$ objects. (Note that for this to exist at all, the domain has to be $\mathsf{FinSet}$, or the codomain has to be $\mathsf{Cat}$). 
This functor $F$ actually happens to have both left and right adjoints (as long as we put in both Fins or neither of them): the left adjoint is the functor which sends a category to its set of connected components, and the right adjoint is the functor which sends a category to its set of objects. But these two functors are not naturally isomorphic, so $F$ is not part of an ambidextrous adjunction. 
