The question:
Let $F:\mbox{Ab} \to \mbox{Ab}$ be an additive functor; if $f$ is a zero homomorphism, then so is $F(f)$; if $A$ is the zero group, then so is $F(A)$.
This boils down to showing that, for an additive functor $F(0)=0$. Obviously this is true, but my category theory is very basic, and I can't quite yet get a handle on what is allowed.
Is it simply just the case that
$F(f+g)=F(f)+f(g) \implies F(0) = F(f+(-f))=F(f)+F(-f)=F(f)-F(f)=0?$