I'm currently reading Paul Cohn's Introduction to Ring Theory, and he states in section 4.5 the following:
A functor $F$ between module categories will be called exact if it transforms any exact sequence $A \stackrel{f}{\to} B \stackrel{g}{\to} C$ into an exact sequence $A^F \stackrel{f^F}{\to} B^F \stackrel{g^F}{\to} C^F$. A sequence $A \stackrel{f}{\to} B \stackrel{g}{\to} C$ is called a null sequence if $g\circ f = 0$, i.e. $\text{im} f \subseteq \text{ker} g$, and it is clear that any additive functor transforms a null sequence into a null sequence.
The last statement here is not at all clear to me and I've tried a few time to show that an additive functor transforms a null sequence into a null sequence, but without much luck. The additive condition on functors doesn't seem that restrictive to me at all, and I'm failing to see how it can be used to deduce information about the image and kernels of its target morphisms.
An additive functor $F:\mathscr C_1\to \mathscr C_2$ induces a group homomorphism from $\text{Mor}_{\mathscr C_1}(A,B)$ to $\text{Mor}_{\mathscr C_2}(FA,FB)$. In particular it maps zero morphisms to zero morphisms. In your example, since $gf=0$ then $g^Ff^F=(gf)^F=0$.