exactness of $A \to B \to C \to 0$

Question

Suppose we want to prove the exactness of the following sequence in an abelian category:

$A \to B \to C \to 0$ and we already have the exactness at $C$.

Suppose now the cokernel $B \to Y$ of $A \to B$ factors through $C$, how to show the exactness at $B$.

I tried using this:Show that in pre-abelian categories, $0 \to A \to B$ is cokernel-exact $\iff$ $A \to B$ is monic but failed.

I even tried using Mod but still cannot see this.

If I am being silly here, please point it out without hesitation. Any help would be appreciated!