Thank you to Achim Krause for pointing out that the first version was broken. Let's try again.

Let $k$ be a finite field. For a set $X$, let $kX$ be the free vector space on $X$. Let $\bigwedge^{\bullet} kX$ be the exterior algebra on $kX$. Then $X \mapsto \bigwedge^{\bullet} kX$ becomes a functor in an obvious way.

Choose any scalars $a_0$, $a_1$, $a_2$, ... in $K$ and define the natural transformation of $\bigwedge^{\bullet} kX$ by multiplying $\bigwedge^{j} kX$ by $a_j$. This gives infinitely (even uncountably) many natural transformations from $X \mapsto \bigwedge^{\bullet} kX$ to itself as a functor from finite sets to finite sets (or even from finite sets to vector spaces).

When studying functors $F$ from the category of finite sets or related categories, one usually wants to impose some sort of finite generation condition, saying roughly that there is some integer $N$ such that any subfunctor of $F$ which agrees with $F$ on sets of size $\leq N$ is the same as $F$. One does this precisely to avoid this sort of trickery with the functor $X \mapsto 2^X$. For example, Eric Ramos, Graham White and I classify functors from FI to FinSet with a finite generation hypothesis and my student John Wiltshire-Gordon classified functors from FinSet to $\mathbb{Q}$-Vect under a similar hypothesis. These are the two papers I know which come closest to studying functors from FinSet to FinSet.

Thank you to Achim Krause for pointing out that the first version was broken. Let's try again.

Let $k$ be a finite field. For a set $X$, let $kX$ be the free vector space on $X$. Let $\bigwedge^{\bullet} kX$ be the exterior algebra on $kX$. Then $X \mapsto \bigwedge^{\bullet} kX$ becomes a functor in an obvious way.

Choose any scalars $a_0$, $a_1$, $a_2$, ... in $k$ and define the natural transformation of $\bigwedge^{\bullet} kX$ by multiplying $\bigwedge^{j} kX$ by $a_j$. This gives infinitely (even uncountably) many natural transformations from $X \mapsto \bigwedge^{\bullet} kX$ to itself as a functor from finite sets to finite sets (or even from finite sets to vector spaces).

When studying functors $F$ from the category of finite sets or related categories, one usually wants to impose some sort of finite generation condition, saying roughly that there is some integer $N$ such that any subfunctor of $F$ which agrees with $F$ on sets of size $\leq N$ is the same as $F$. One does this precisely to avoid this sort of trickery with the functor $X \mapsto 2^X$. For example, Eric Ramos, Graham White and I classify functors from FI to FinSet with a finite generation hypothesis and my student John Wiltshire-Gordon classified functors from FinSet to $\mathbb{Q}$-Vect under a similar hypothesis. These are the two papers I know which come closest to studying functors from FinSet to FinSet.

Let $F=G$ be the functor with $F(X)$ equal to set of the subsets of $X$, and, for $f: X \to Y$ and $S \subseteq X$, we put $F(f)(S) = f(S)$.

  Let $k$ be a positive integer. We define a natural transformation $\phi_k : F \to G$ as follows: $$\phi_k(S) = \begin{cases} S & |S| \geq k \\ \emptyset & |S| < k \end{cases}.$$

Clearly, all the $\phi_k$ are distinct natural transformations.

When studying functors $F$ from the category of finite sets or related categories, one usually wants to impose some sort of finite generation condition, saying roughly that there is some integer $N$ such that any subfunctor of $F$ which agrees with $F$ on sets of size $\leq N$ is the same as $F$. One does this precisely to avoid this sort of trickery with the functor $X \mapsto 2^X$. For example, Eric Ramos, Graham White and I classify functors from FI to FinSet with a finite generation hypothesis and my student John Wiltshire-Gordon classified functors from FinSet to $\mathbb{Q}$-Vect under a similar hypothesis. These are the two papers I know which come closest to studying functors from FinSet to FinSet.

Thank you to Achim Krause for pointing out that the first version was broken. Let's try again.

Let $k$ be a finite field. For a set $X$, let $kX$ be the free vector space on $X$. Let $\bigwedge^{\bullet} kX$ be the exterior algebra on $kX$. Then $X \mapsto \bigwedge^{\bullet} kX$ becomes a functor in an obvious way.

Choose any scalars $a_0$, $a_1$, $a_2$, ... in $K$ and define the natural transformation of $\bigwedge^{\bullet} kX$ by multiplying $\bigwedge^{j} kX$ by $a_j$. This gives infinitely (even uncountably) many natural transformations from $X \mapsto \bigwedge^{\bullet} kX$ to itself as a functor from finite sets to finite sets (or even from finite sets to vector spaces).

When studying functors $F$ from the category of finite sets or related categories, one usually wants to impose some sort of finite generation condition, saying roughly that there is some integer $N$ such that any subfunctor of $F$ which agrees with $F$ on sets of size $\leq N$ is the same as $F$. One does this precisely to avoid this sort of trickery with the functor $X \mapsto 2^X$. For example, Eric Ramos, Graham White and I classify functors from FI to FinSet with a finite generation hypothesis and my student John Wiltshire-Gordon classified functors from FinSet to $\mathbb{Q}$-Vect under a similar hypothesis. These are the two papers I know which come closest to studying functors from FinSet to FinSet.