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.

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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][1] classify functors from FI to FinSet with a finite generation hypothesis and my student [John Wiltshire-Gordon][2] 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.


  [1]: https://arxiv.org/abs/1804.04238
  [2]: https://arxiv.org/abs/1406.0786