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 $?
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 $?
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).
Consider $\mathrm{Set}^{\mathrm{M}^{\operatorname{op}}}$, where $M$ is a group viewed as a category. This is equivalent to the category of $\mathrm{Set-M}$ of right $M$ actions. If you write out the commutative diagram for a natural transformation for a functor $F\colon \mathrm{M}^{\operatorname{op}} \to \mathrm{Set}$ you see that you only need a non-identity map of right M-Sets between $X$ and itself ($X$ is the image of the only object in the category $\mathrm{M}$), which exists in general.