Let $\mathcal{C}$ be a category, and suppose for all objects $X, Y$ of $\mathcal{C}$, Mor$_{\mathcal{C}}(X,Y)$ is equipped with the structure of an Abelian group, such that the composition of morphisms is bilinear.
Now it can easily be shown that Mor$_{\mathcal{C}}(X,X) =:$ End$_{\mathcal{C}}(X)$ is a ring (under composition as multiplication). (In the absence of any knowledge about the Abelian group structure, distributivity still arises from bilinearity in this case.)
The next thing to be proved is that $X$ is a "zero object" if and only if End$_{\mathcal{C}}(X)$ is the zero ring. Now I see that this "zero object" takes the form of a trivial group in group theory, zero ring in ring theory, the space $\{0\}$ as a vector space, etc., but I haven't come across a clear general form of this concept, which makes it hard to prove the statement for a general object $X$ of this rather general category $\mathcal{C}$.
A zero object $X$ is defined to be both a terminal and an initial object, i.e. there is exactly one arrow $A\to X$ and exactly one arrow $X\to A$.
Since each $\hom(A,B)$ is an Abelian group, it always has a zero element, call it $0_{AB}$.
Note that bilinearity implies $0_{AB}\circ f=0_{XB}$ for any $f:X\to A$.
Now, if $X$ is a zero object, also $\hom(X,X) $ has only one element, hence $1_X=0_{XX}$, so it must be the zero ring.
Conversely, if $\hom(X,X)$ is the zero ring, we have $1_X=0_{XX}$ and thus, for any $f:A\to X$, we have $$f=1_X\circ f=0_{XX}\circ f=0_{AX}$$ So there's a unique arrow $A\to X$.
And similarly, $0_{XA}$ is the unique arrow $X\to A$.