Is the powerset functor self-adjoint?

Question

Let $F$ denote the power-set functor. That is, $F:Set\rightarrow Set$ by taking sets to their power-sets and if $f:X\rightarrow Y$ then $F(f)$ takes subsets of $X$ to their image.

Is $F$ self-adjoint?

Answer 1

Suppose that a set $X$ has $m$ elements and that a set $Y$ has $n$ elements ($m,n\in\mathbb N$). Then$$\#\hom\bigl(\mathcal{P}(X),Y\bigr)=n^{2^m}\text{ and }\#\hom\bigl(X,\mathcal{P}(Y)\bigr)=(2^n)^m=2^{mn}.$$Since, in general, $n^{2^m}\neq2^{mn}$…