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creatd connecting homomorphism with (just) the pedestrian description.
(Relation to snake lemma and more generally to fiber sequences not there yetâ€¦)
There is an influential axiomatics of such homomorphisms in Tohoku, where Grothendieck talks about $\delta$-functors, and dually, $\delta^*$-functors, regardless their origin or construction. The connecting homomorphism is just part of the data for such a functor (which deserves a $n$lab entry, but I am in troubles not allowing me to contribute much to $n$lab at the moment). Alexander Rosenberg generalized this axiomatics for nonabelian complexes with objects in what he calls right exact categories, i.e. categories with choice of subcanonical singleton pretopology by strict epimorphisms.
added to connecting homomorphism the diagrammatic proof in an arbitrary abelian category
added now to connecting homomorphism also an explicit element-chasing proof.
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