I am trying to prove whether in a pre-additive category, $0_{Mor(y,z)}\circ f=0_{Mor(x,z)}$ for objects $x,y,z$ and $f\in Mor(x,y)$. Now, by bi-linearity of composition maps $$ 0_{Mor(y,z)}\circ f+ 0_{Mor(y,z)}\circ (-f)= 0_{Mor(y,z)} \circ 0_{Mor(x,y)} $$
Putting $f=0_{Mor(x,y)}\; \;$ we have $\; \; 0_{Mor(y,z)} \circ 0_{Mor(x,y)} =0_{Mor(x,z)}\; \; $and we also get $\; \; 0_{Mor(y,z)}\circ f=-( 0_{Mor(y,z)}\circ (-f))$. But I can't seem to progress any further.
Thanks in advance!
The answer to your question is basically in asdq's comment, allow me to expand a little.
By bilinearity you have that for $f \in Mor(x,y)$ the mapping $$ \begin{align*} -\circ f \colon Mor(y,z) &\to Mor(x,z) \\ x \mapsto x \circ f \end{align*} $$ is a group homomorphism, in particular it sends $0_{Mor(y,z)}$ into $0_{Mor(x,z)}$, that is that $0_{Mor(y,z)}\circ f = 0_{Mor(x,z)}$.