TITLE: Existence of non-identity natural transformations $\tau:F \to F $
QUESTION [2 upvotes]: Consider a functor $F:A\to B$. It is an object in category $B^A$. 
Question: is $B^A(F,F)$ a singleton? 
To put is in other words: do we have non-identity natural transformations $\tau:F \to F $?

REPLY [4 votes]: I hope Andy won't be bothered if I answer witht he "general case" of his construction. Take any category $\mathbf C$ and the hom-functor $F=\hom(X,-)$; then by Yoneda Lemma 
$$
Nat(F,F)\cong \hom(X,X)
$$
Now you just have to choose "properly" your $X$ (e.g. by taking an object with a non-trivial endomorphism monoid).
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