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

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

[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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