Let $C,D$ be abelian categories and let $F:C\rightarrow D$ be an exact functor. I wonder if is it true that if $A$ is a finite length object in $C$ then $F(A)$ is finite length in $D$.
Since we assume that $F$ is exact, the question reduces to prove or disprove that if $A$ is a simple object then $F(A)$ is finite length object.
In case it is not true, I would love to see a counterexample for that.
Thanks.