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Let $\alpha : A \xrightarrow{} B$ be a map of chain complexes. If $\alpha$ is chain homotopic to the zero map via a homotopy $h$, it is well known that the map $h$ extends to a chain map $M(\alpha) \xrightarrow{} B$.

My question is, for two different null homotopies , are the two resulting extensions $M(\alpha) \xrightarrow{} B$ chain homotopy equivalent?

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