Slightly simplified, the five lemma states that if we have a commutative diagram (in, say, an abelian category)

$$\require{AMScd} \begin{CD} A_1 @>>> A_2 @>>> A_3 @>>> A_4 @>>> A_5\\ @VVV @VVV @VVV @VVV @VVV\\ B_1 @>>> B_2 @>>> B_3 @>>> B_4 @>>> B_5 \end{CD} $$ where the rows are exact and the maps $A_i \to B_i$ are isomorphisms for $i=1,2,4,5$, then the middle map $A_3\to B_3$ is an isomorphism as well.

This lemma has been presented to me several times in slightly different contexts, yet the proof has always been the same technical diagram chase and no further intuition behind the statement was provided. So my question is: do you have some intuition when thinking about the five lemma? For instance, particular choices of the $A_i, B_i$ which make it more transparent why the result should be true? Some analogy, heuristic, ...?

`soft-question`

to a bit of extreme. Good answers to your question will likely provide some computations and examples of homology, and perhaps quite hard computations, the subject being homological algebra. So (as the author of the tag ;)) I think I'll do some retag. – Ilya Nikokoshev Oct 26 '09 at 23:17