The target of an epimorphism which is also a zero morphism is a zero object.

Question

I'm stuck on the following problem.

In a category with a zero object; if an epimorphism is a zero morphism, then its target is a zero object.

Answer 1

Note that in a cat. with zero object, any initial object is the zero object (because all initial objects are isomorphic).

Thus, it is enough to show that the codomain $y$ of the said zero epimorphism $x \stackrel{0}\twoheadrightarrow y$ is initial. So given $z$, pick any two morphisms $\alpha, \beta: y \rightarrow z$ (also show that there is at least one such morphism). Now show that $\alpha$ is necessarily equal to $\beta$.