Consequence of five or four lemma?

Question

I have two horizontal exact sequences of abelian groups.

$\require{AMScd}$ \begin{CD} 0 @>{}>> A_1 @>{}>> A_2 @>{}>> A_3 @>{}>> A_4 @>{}>> 0\\ @| @VeVV @VfVV @VgVV @VhVV @|\\ 0 @>{}>> B_1 @>{}>> B_2 @>{}>> B_3 @>{}>> B_4 @>{}>> 0 \end{CD}

where $f,g$ And $h$ are isomorphisms. By diagram chasing I can figure out that $e$ is also an isomorphism. But is this also a consequence of the five or four lemma?

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I forgot to mention that the squares commute

Answer 1

You can extend your diagram on the left side by two zeroes to get the typical five lemma situation where the morphism $e$ is in the middle and will be an isomorphism as $f$ and $g$ are isomorphisms (and the morphisms between the zeroes as well): $\require{AMScd}$ \begin{CD} 0 @>>> 0 @>>> A_1 @>>> A_2 @>>> A_3 @>>> A_4 @>>> 0 \\ @V0VV @V0VV @VeVV @VfVV @VgVV @VhVV @V0VV \\ 0 @>>> 0 @>>> B_1 @>>> B_2 @>>> B_3 @>>> B_4 @>>> 0 \\ \end{CD}