I’ve been struggling with this for some time. I can prove the Snake Lemma, but I don’t really “understand” it. By that I mean if no one told me Snake Lemma existed, I would not even attempt to prove it. I believe I am missing some important intuition that tells me that the lemma “could” be true before actually trying to prove it. Please give me some pointers.
All these diagram chasing lemmas (snake lemma, 3x3 lemma, four lemma, five lemma, etc.) follow directly from the “salamander lemma” due to George Bergman, see salamander lemma.
And that is pretty transparent. It is so transparent that for instance it is immediate to see (which no textbook ever mentions) that there are just as easily 4×4 lemmas, 5×5 lemmas, 6×6 lemmas. etc. In other words: once you see the simple idea of the salamander lemma, you can come up yourself with more such diagram chasing lemmas at will.