Zero object equivalent assertion

Question

Let C be a category with zero object $0$.

(i) Prove that for $A \in C$ the following assertions are equivalent:

(a) A is a zero object;
(b) $id_A$ is a zero morphism;
(c) there is a monomorphism $A \to 0$;
(d) there is an epimorphism $0 \to A$.

I need help to prove c implies d.

Progress: I reduced the problem to prove that $A$ is an initial object and using that f is monic. I'm already solve this problem. Thanks.

Answer 1

For every object $ A $, the arrow $ A\longrightarrow 0$ is a split epimorphism. Therefore (c) says that this arrow is an isomorphism, which implies (d).