I'm trying to understand what the snake lemma 'computes' on small examples. Consider this: 
It seems to me that given two 'defective' / 'incomplete' simplicial complexes that are related through a chain morphism, the snake lemma tells us 'how best to combine them'. In this case for example, it tell us to send $v$ to $f'$ which I interpret to mean 'glue $f'$ to $v$ in the correct way [mediated by $e$].
- Is this interpretation of the use of the snake lemma correct?
- If it is, then what happens in other situations of using the snake lemma? I am looking for an answer couched in terms of geometry or simplicial homology [one that can be drawn / visualized]
- If my interpretation is wrong, then what's the correct interpretation in this case?