Lemma 12.5.13. Let $\mathcal{A}$ be an abelian category. Let

$\xymatrix{ w\ar[r]^ f\ar[d]_ g & y\ar[d]^ h\\ x\ar[r]^ k & z }$

be a commutative diagram.

1. If the diagram is cartesian and $k$ is an epimorphism, then the diagram is cocartesian and $f$ is an epimorphism.

2. If the diagram is cocartesian and $g$ is a monomorphism, then the diagram is cartesian and $h$ is a monomorphism.

Proof. Suppose the diagram is cartesian and $k$ is an epimorphism. Let $u = (g, f) : w \to x \oplus y$ and let $v = (k, -h) : x \oplus y \to z$. As $k$ is an epimorphism, $v$ is an epimorphism, too. Therefore and by Lemma 12.5.11, the sequence $0\to w\overset {u}\to x\oplus y\overset {v}\to z\to 0$ is exact. Thus, the diagram is cocartesian by Lemma 12.5.11. Finally, $f$ is an epimorphism by Lemma 12.5.12 and Lemma 12.5.4. This proves (1), and (2) follows by duality. $\square$

Comment #719 by Anfang Zhou on

Topy. I think we should either change $u=(g,-f)$ to be $u=(g,f)$ or change $v=(k,-h)$ to be $v=(k,h)$.

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