Is this an instance of the snake lemma?

Question

I recently had need of the following fact (in the category of abelian groups, but I'm pretty sure it holds for all abelian categories): given a commutative diagram of the form

(quiver link), thus $k \circ g = l \circ h$, $f$ and $g$ are epimorphisms, and $l$ is a monomorphism, there is a canonical long exact sequence $$ 0 \to \ker f \to \ker g \circ f \to \ker h \to \ker k \to 0$$ defined in a natural fashion. Verifying that this fact is a routine (but tedious) diagram chase. Given the similarity to the snake lemma, I was initially certain that this fact was somehow a corollary of the snake lemma, but I was only able to use the snake lemma to establish exactness at $\ker k$ and nowhere else. Can the entirety of this sequence be established from the snake lemma, or one of the other canonical diagram chase lemmas?

EDIT: As noted in comments, the claim follows in fact from splicing together two applications of the snake lemma, namely

and

Answer 1

Splice the left ends of the kernel-cokernel exact sequences $0 \to \ker(f) \to \ker(gf) \to \ker(g) \to \mathrm{cok}(f) \to \mathrm{cok}(gf) \to \mathrm{cok}(g) \to 0$ and $0 \to \ker(g) \to \ker(kg) \to \ker(k) \to \mathrm{cok}(g) \to \mathrm{cok}(kg) \to \mathrm{cok}(k) \to 0$, noting that $\ker(kg) = \ker(h)$, $\mathrm{cok}(f) = 0$ and $\mathrm{cok}(g) = 0$. The kernel-cokernel sequence for a composition appears in Milne's Arithmetic Duality Theorems (Proposition 0.24), Short exact sequences every mathematician should know and https://arxiv.org/abs/2001.07528 , but must be about as old as the snake lemma.