I'm working on the following question:
"Let $A_{*}$ be a chain complex. We have the cylinder chain complex $A^c_*$
$ ...\xrightarrow{\partial_3^c} A_1\oplus (A_2\oplus A_2) \xrightarrow{\partial_2^c} A_0\oplus (A_1\oplus A_1)\xrightarrow{\partial_1^c} A_{-1}\oplus (A_0\oplus A_0)\xrightarrow{\partial_0^c}... $
with degree $n$ term $A_{n-1} \oplus (A_n \oplus A_n)$ and boundary maps
$\partial_n^c(\sigma,(\tau_1,\tau_2)):=(\partial\sigma,((-1)^{n+1} \sigma+\partial\tau_1,(-1)^n+\partial\tau_2))$
Show that this defines a chain complex. Show that a chain homotopy of two chain maps from $A_*$ to $B_*$ may be identified with a chain map $h_\sharp:A^c_*\rightarrow B_*$."
I found it easy to show that this defines a chain complex, but couldn't find a chain map $h_\sharp$ based off a chain homotopy of two chain maps from $A_*$to $B_*$ that would commute.
I tried simply assuming $\bar{h}:A_n\rightarrow B_{n+1}$ was the chain homotopy of the two chain maps, and defining $h_n(\sigma,(\tau_1,\tau_2)):=\bar{h}_{n-1}(\sigma)$, but I got $\partial_{n}^Bh_n\neq h_{n-1}\partial_{n}^C$.