Endofunctor on abelian category which fixes simples

Let $\mathcal{A}$ be an abelian category enriched over $\mathbb{C}$ such that every object in $\mathcal{A}$ has finite length. Suppose that $F:\mathcal{A}\to\mathcal{A}$ is an exact endofunctor such that for every simple object $L$ in $\mathcal{A}$ we have an isomorphism $F(L)\cong L$.

Then does this imply that $F$ is naturally isomorphic to the identity functor on $\mathcal{A}$?

Solution 1:

I don’t think so. Consider for $\mathcal{A}$ the category of the $(V,\sigma)$ where $V$ is a finite-dimensional $\mathbb{C}$-vector space and $\sigma$ is a nilpotent endomorphism. Its only simple object is $S=(\mathbb{C},0)$.

Let $F$ be the “semi-simplification” functor $(V,\sigma) \longmapsto (V,0)$, and mapping $f:(V,\sigma) \rightarrow (W,\tau)$ to $f: (V,0) \rightarrow (W,0)$. Then $F(S)$ is isomorphic to $S$. But $F$ is not naturally isomorphic to the identity functor.