On the Wikipedia page for zero morphisms it states that
any morphism composed with a zero morphism gives a zero morphism.
But I'm not sure how to prove this fact? It seems obvious that when you are composing two zero morphisms that the result is a zero morphism (by commutativity of two triangles); but what if you are composing a zero morphisms with a non-zero morphism?
A morphism $f\colon A\to B$ is a zero morphism if it factors through a zero object $Z$, that is, $f=f_2f_1$, with $f_1\colon A\to Z$ and $f_2\colon Z\to B$.
A zero object $Z$ is characterized (up to unique isomorphism) by the fact that, for every object $X$, there is a unique morphism $Z\to X$ and a unique morphism $X\to Z$ (that is, it is both initial and terminal).
Note that the factorization $f_2f_1$ is unique, by definition of zero object, once a particular zero object has been chosen.
If $g\colon B\to C$, then $gf$ factors through $Z$ as well. Similarly if composition is on the other side.