Let $f:B\to C$ be a morphism of chain complexes. Exercise 1.5.9 from Weibel's "An introduction to homological algebra", page 24, states that the natural maps $\ker f[-1]\xrightarrow\alpha{\rm cone}(f)\xrightarrow\beta{\rm coker} f$ give rise to a long exact sequence: $$\cdots\xrightarrow\delta H_{n-1}(\ker f)\xrightarrow\alpha H_n({\rm cone}(f))\xrightarrow\beta H_n({\rm coker} f)\xrightarrow\delta H_{n-2}(\ker f)\xrightarrow\alpha\cdots$$ (In the book there is a small mistake. The map $\ker f[-1]\to{\rm cone}(f)$ was denoted by $\delta$ instead of $\alpha$.)

My question is, is there any quotable source for this result? I cannot quote an exercise, especially when it is incomplete. It doesn't state what the map $\delta$ is and in my paper I need that $\delta$.

I figured out that $\delta=\delta'\delta''$, where $\delta'$ is the connecting map $H_{n-1}({\rm Im}f)\to H_{n-2}(\ker f)$, coming from the exact sequence $0\to\ker f\to B\xrightarrow f{\rm Im}f\to 0$, and $\delta''$ is the connecting map $H_n({\rm coker}f)\to H_{n-1}({\rm Im}f)$, coming from the exact sequence $0\to{\rm Im}f\to C\to{\rm coker} f\to 0.$

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