Let $\mathcal{C}$ be a preadditive category, i.e. each hom-set admits an abelian group structure and composition with respect to these structures is bilinear. There is the well-known fact that if finite products exist, then so fo finite coproducts and they coincide. I am having some trouble proving the zero-ary case:
If $\mathcal{C}$ has an initial object $\varnothing$, then $\varnothing$ is terminal.
We have to show that for every object $X \in \mathcal{C}$ there exists a unique arrow $X \to \varnothing$. Existence is easy. Since each hom-set is a group, we simply let $0 : X \to \varnothing$ be the zero element of the group $\mathcal{C}(X,\varnothing)$. However, uniqueness is bothering me. I think there is not much choice: We suppose there exists another morphism $f : X \to \varnothing$. Since $\varnothing$ is initial, we find $h : \varnothing \to X$. Then since $\mathcal{C}(\varnothing,\varnothing)$ has only one element, we get that $$0 \circ h = f \circ h = \operatorname{id}_\varnothing$$ But then I am stucked...any hint or alternative way would be nice (I am aware of the proof found here, but somehow I would like to keep my phrasing).