Let $E : \mathbf{Set} \to \mathbf{Set}$ be the contravariant functor taking a set $X$ to the set $E(X)$ of distinct equivalence relations on $X$. It takes (I assume) a function $f:X \to Y$ to a function $f^*:E(Y) \to E(X)$ sending each equivalence relation on $Y$ to the relation on $X$ defined by $x_1 \sim x_2$ whenever $f(x_1) \sim f(x_2)$. Is this functor representable?
This is an exercise (9.2.3) from the text "An Invitation to General Algebra and Universal Constructions" by Bergman. In the back of the book, he gives an extremely terse hint that seems to imply it is representable.
I'm thinking it's not representable. A representing object $R$ should have $\mathrm{Hom}(-,R) \cong E$. For a finite set $X$, $E(X)$ is finite, so $\mathrm{Hom}(X,R)$ is finite, so $R$ must be a finite set. But then, for finite sets $X$, $\mathrm{Hom}(X,R)$ has cardinality $|R|^{|X|}$, while $E(X)$ is the number of partitions on $X$. There's no number $|R|$ making these values coincide, so I conclude $E$ is not representable. Is there a mistake in my reasoning?