Any reference for induces exact sequence $0 \to Ker(f) \to Ker(gf) \to Ker(g) \xrightarrow{\delta} Coker(f) \to Coker(gf) \to Coker(g) \to 0.$

Question

Let $\mathscr{C}$ be an abelian category and $f:A \to B$ and $g:B \to C$ morphisms in $\mathscr{C}$, then we have the following exact sequence

$$0 \to \operatorname{Ker}(f) \to \operatorname{Ker}(gf) \to \operatorname{Ker}(g) \xrightarrow{\delta} \operatorname{Coker}(f) \to \operatorname{Coker}(gf) \to \operatorname{Coker}(g) \to 0.$$

I need to use this result which I know its true since I almost got it, but instead of looking help ending the proof. I'm wondering if someone knows a book or text where I can reference this result? This one looks pretty much like the Snake Lemma but I cannot find them in Literature. Or if someone help me see this as an application of a the Snake Lemma it would also be helpful. Thanks

Answer 1

It is a consequence of the snake lemma applied to $$\array{ \operatorname{Ker}(f) &\to & A &\stackrel{f}{\to} & B &\to& \operatorname{Coker}(f) &\to & 0 \\ && \downarrow^{gf} && \downarrow^{g} && \downarrow \\ 0 &\to& C &\to & C &\to& 0 } $$ and you get $$ 0 \to \operatorname{Ker}(f) \to \operatorname{Ker}(gf) \to \operatorname{Ker}(g) \to \operatorname{Coker}(f) \to \operatorname{Coker}(gf) \to \operatorname{Coker}(g) \to 0$$

Answer 2

Try applying the snake lemma to the diagram $$ \require{AMScd} \begin{CD} 0 @>>> A @>{\begin{bmatrix} id \\ f \end{bmatrix}}>> A \oplus B @>{\begin{bmatrix} -f & id \end{bmatrix}}>> B @>>> 0 \\ @. @VVfV @VV{\begin{bmatrix} 0 & id \\ gf & 0 \end{bmatrix}}V @VV{-g}V \\ 0 @>>> B @>{\begin{bmatrix} id \\ g \end{bmatrix}}>> B \oplus C @>{\begin{bmatrix} -g & id \end{bmatrix}}>> C @>>> 0. \end{CD} $$