[Lemma 4.5 page 16, Eilenberg-Steenrod]

Lemma 12.5.20. Let $\mathcal{A}$ be an abelian category. Let

$\xymatrix{ v \ar[r] \ar[d]^\alpha & w \ar[r] \ar[d]^\beta & x \ar[r] \ar[d]^\gamma & y \ar[r] \ar[d]^\delta & z \ar[d]^\epsilon \\ v' \ar[r] & w' \ar[r] & x' \ar[r] & y' \ar[r] & z' }$

be a commutative diagram with exact rows. If $\beta , \delta$ are isomorphisms, $\epsilon$ is injective, and $\alpha$ is surjective then $\gamma$ is an isomorphism.

Proof. Immediate consequence of Lemma 12.5.19. $\square$

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