I am currently working on problem 9.4.10 in Aluffi and the problem states: Prove that two cochain maps $\alpha^*, \beta^*: L^* \rightarrow M^*$ are homotopic (There exist certain maps $h^i: L^i \rightarrow M^i$ such that $\beta^i - \alpha^i = d_m^{i-1} \circ h^i + h^{i+1} \circ d_L^i$) iff they extend to a cochain morphism $(-\alpha^*, h^*, \beta^*): MCyl(id_{L^*}) \rightarrow M^*$.
We defined $MCyl(\alpha)^*$ to be a cochain complex with objects $L^i \oplus L^{i+1} \oplus M^i$. I have two questions: How is $MCyl(id_{L^*})$ defined since the identity is not a map from $L^* \rightarrow M^*$? In the ordered triple of maps above, how does $\beta^*$ go from $M^* \rightarrow M^*$?
Here is my answer: $MCyl(id_{L^*})$ has objects $L^i \oplus L^{i+1} \oplus L^i$.