Mathematical proofs based on chasing elements in commutative diagrams

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### Are these connecting homomorphisms commutative?

Are the connecting homomorphism induced by Kummer sequence and that of localization sequence commutative?
In other words, is the following statement true?
If it is true, then, how can one prove it?
...

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### Five-lemma for the end of long exact sequences of homotopy groups

Consider the commutative diagram below with exact rows (from the long exact sequence of homotopy groups) and $f_1,f_2,f_4,f_5$ bijective ($f_1,f_2$ homomorphisms). Does it follow that $f_3$ is also ...

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### Is this isomorphism canonical?

Suppose $A\leq A',B$ and $C' \leq C$ are (finite dimensional) vector spaces.
Suppose that
$$ 0 \to A \to B \to C \to 0 $$
$$ 0 \to A' \to B \to C' \to 0 $$
are exact. Then using a dimension argument ...

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### What are the oldest illustrations of “Venn” diagrams?

Graphical representations of intersection of sets as logical combinations are much older than Venn.
Euler and Leibniz are often quoted and the current Wikipedia article also quotes Ramon Llull but I ...

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### When does adding inverses of morphisms preserve commutativity of a diagram?

Here is the essence of a problem I have run in to: I have a finite poset D with a terminal object. If I formally invert all of the morphisms, and add these into my diagram, does the new diagram D' ...

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### The sharp 3x3 lemma: a proof by universal properties?

I was reading this paper a while ago, and I couldn't figure out how to prove a lemma that was left as an exercise by only using universal properties and the definition of an abelian category.
I'll ...