normal epimorphisms

Question

In a category $\mathcal{C}$ consider the diagram $$K\longrightarrow A\longrightarrow B$$ with $k:K\longrightarrow A$ the kernel of $f:A\longrightarrow B$ and $f$ a cokernel. Is it true that $f$ is the cokernel of $k$?

Answer 1

Let $f$ be a cokernel of $g\colon C\to A$. Since $fg = 0$ and $k$ is kernel of $f$, there exists unique $\bar g\colon C\to K$ such that $g = k\bar g$. Now, let $h\colon A\to D$ such that $hk = 0$. This implies that $0 =hk\bar g = hg$. But $f$ is cokernel of $g$, so there exists unique $\bar h\colon B\to D$ such that $h = \bar h f$. Thus, $f$ is cokernel of $k$.