If $0$ is the zero-object $ \Longrightarrow F(0) $ is the zero object when $F$ additive

Question

Let $$ F : \text{A-Mod} \to \text{A-mod} $$ be an additive functor. Then if $0$ is the zero-object $F(0) $ is the zero object. Why this is true ?

The definition of additive functor that I know is

$\forall \ \ M , N \in $ A-Mod $$F : \hom_{A-Mod}(M,N ) \to \hom_{A-Mod}(FM,FN ) $$ is a morphism of abelian groups.

Answer 1

Let $0_M$ and ${\rm id}_M$ denote the zero map and identity map on an $A$-module $M$. We have

$$M=0\iff 0_M={\rm id}_M.$$

Since $F$ is a functor, $F({\rm id}_M)={\rm id}_{FM}$. Since it's also additive, $F(0_M)=0_{FM}$.