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There is something which really bothers me. In all the references I looked (including the internet), no one talks about the uniqueness of the connecting morphism constructed in the Snake Lemma. The only source I found is this (beautiful) website : Uniqueness of the connecting morphism where it is made clear that we do not have the strict uniqueness.

My question is then : Is the connecting morphism $\delta : Ker \ c \rightarrow Coker \ a$ unique up to isomorphism ? I'm working with abelian categories, so I would appreciate a "purely" categorical argument.

Sov
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  • I'm just going to leave this here. – Xander Henderson Apr 12 '18 at 16:30
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    You can find a "purely categorical" proof (by which I mean a proof without diagram chasing, although this is perfectly valid in abelian categories as well) in Categories and Sheaves at page 297. There the connecting morphism is constructed via universal properties, which immediately implies uniqueness. – asdq Apr 12 '18 at 16:35
  • What do you mean by unique up to isomorphism ? Of course, if you have a fix diagram, the snake lemma says that we have an isomorphism $\operatorname{coker}(\ker b\to\ker c)\simeq \ker(\operatorname{coker}a\to\operatorname{coker}b)$, if there is one, there usually plenty of them... As an example, you can always take $-\delta$. Now, the snake lemma produces a canonical one (and natural). – Roland Apr 12 '18 at 19:36
  • Isn't this a duplicate of the question https://math.stackexchange.com/questions/1491307/ you linked yourself? – Ben Apr 13 '18 at 08:58

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The difficulty is in finding the property with respect to which the connecting morphism should be unique. A complete and purely categorical answer is given in this result in the Stacks Project.

Fred Rohrer
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