I seem to remember a proof that if a category $C$ has coproducts and a $0$ object, then necessarily if we had objects of $C$, say $a$ and $-a$, such that $a \oplus -a \simeq 0$, then $a\simeq 0\simeq -a$. 


But right now, I can't place this, nor am I 100% sure that that is the correct property. I am able to show that, using the various universal properties at play, that the morphisms in such a category are necessarily quite boring, but not that it completely collapses (i.e. if I assume that all objects have a corresponding 'negative' object).



Negative objects in categories
With Yoneda ? For every object $X$, $Mor(a\oplus (-a),X)$ is a singleton since $a \oplus (-a)$ is initial. And $Mor(a\oplus (-a),X) \cong Mor(a,X) \times Mor((-a),X)$ is a singleton as well. So $Mor(a,X) \cong Mor((-a),X) \cong Mor(0,X)$. Hence $a \cong (-a) \cong 0$. 


