## How to prove that “equivalences” of complexes of sheaves define an equivalence relation?

An equivalence of two complexes $K^{\cdot}$ and $L^{\cdot}$ of sheaves (of abelian groups) on a spaces $X$ is a triple $(M^{\cdot}, \alpha,\beta)$ where $M^{\cdot}$ is a complex of sheaves on $X$ and $\alpha:K^{\cdot}\to M^{\cdot}$ and $\beta:L^{\cdot}\to M^{\cdot}$ are quasi-isomorphism.

In order to show the equivalences define indeed an equivalence relation between complexes, one need to show the following lemma: "Suppose quasi-isomorphism $\alpha':N^{\cdot}\to K^{\cdot}$ and $\beta': N^{\cdot}\to L^{\cdot}$ are given. Then the complex $M^{\cdot}:=(K^{\cdot}\oplus L^{\cdot})/(\alpha',\beta')N^{\cdot}$ together with the natural maps from $K^{\cdot}$ and $L^{\cdot}$ form an equivalence of $K^{\cdot}$ and $L^{\cdot}$."

However, I can't claim the above lemma is true.

Reference: MHS and singularities, page35 (a lecture from a seminar in department of mathematics in Mainz University, 2008).

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Construct an exact sequence with $M$ in the middle and use the Five Lemma. – Martin Brandenburg Jul 12 2011 at 11:53