There is more than one functor that fix objects in category of groups? I am trying to solve the following exercise:
Find two functors $F_1,\ F_2: Grp \rightarrow Grp $ such that $F_i G = G $ for all $i=1,2$ and $G \in Grp$, where $Grp$ is the category of groups. 
Of course the identity functor can be chosen like one of this $F_i's$ but I cant find another. 
Thank's
 A: Let $\alpha$ denote a function that assigns to each group $G$ an automorphism $\alpha_G : G \leftarrow G$. Then we get a corresponding identity-on-objects functor $\mathbf{f}_\alpha : \mathbf{Grp} \leftarrow \mathbf{Grp}$ by asserting that for every morphism $\varphi:H \leftarrow G$, we have: $$\mathbf{f}_\alpha(\varphi) = \alpha_H \circ \varphi \circ \alpha^{-1}_G.$$
If $\alpha$ is defined such that for all groups $G$ we have $\alpha_G = \mathrm{id}_G,$ then $\mathbf{f}_\alpha$ is just $\mathrm{id}_\mathbf{Grp}$. But for other choices of $\alpha$, we can potentially get other functors $\mathbf{Grp} \leftarrow \mathbf{Grp}$ that fix all objects. This really has nothing to do with $\mathbf{Grp}$, of course; it works for any category whatsoever. Neat exercise, by the way!
Anyway, lets be a little more explicit. Let $K$ denote your favourite group and $k$ denote a non-central automorphism of $K$. Let $\alpha$ assign to every group the identity function, except for $K$ which is assigned the automorphism $k$. Then $\mathbf{f}_\alpha$ is an identity-on-objects functor that is distinct from $\mathbf{id}_\mathbf{Grp}$. To see this, let $j$ denote an automorphism of $K$ that doesn't commute with $k$. So $j \circ k \neq k \circ j$. Hence $j \neq k \circ j \circ k^{-1}.$ In other words, $j \neq f_\alpha(j)$.
