I am studying the construction of Steenrod squares from Bredon's book "Topology and Geometry", page 413. I use $C_*(X)$ to denote the singular chain complex, while Bredon uses $\Delta_*(X)$.
He starts the construction by considering a diagonal approximation $\Delta:C_*(X)\to C_*(X)\otimes C_*(X)$ and the chain map $T:C_*(X)\otimes C_*(X)\to C_*(X)\otimes C_*(X)$ with $T(\sigma^p\otimes\sigma^q)=(-1)^{pq}\sigma^q\otimes\sigma^p$. Then $T\circ\Delta_0$ is again a diagonal approximation thus there exists a chain homotopy $\Delta_1:C_*(X)\to C_*(X)\otimes C_*(X)$, which is a map of degree 1, such that $T\Delta_0-\Delta_0=\partial\Delta_1+\Delta_1\partial$, where the first $\partial=\partial^{\otimes}$ is the boundary map for $C_*(X)\otimes C_*(X)$ and the second for $C_*(X)$. Note that $T^2-1=0$ so composing with $(T+1)$ the previous relation we get $\partial[(T+1)\Delta_1]+[(T+1)\Delta_1]\partial=(T+1)[\partial\Delta_1+\Delta_1\partial]=(T+1)(T-1)=0$. This means that $(T+1)\Delta_1$ is a natural chain map of degree 1. Another such map is the zero mapping so by the method of acyclic models there exists a natural chain homotopy $\Delta_2$ between them. That is $(T+1)\Delta_1=\partial\Delta_2-\Delta_2\partial$.
Then inductively we can construct chain homotopies $\Delta_n$ of degree $n$ such that $(T+(-1)^{n+1})\Delta_n=\partial\Delta_{n+1}+(-1)^n\Delta_{n+1}\partial$. Then he dualizes and defines the cup-i product.
Here is my problem. I think it should be $(T+1)\Delta_1=\Delta_2\partial-\partial\Delta_2$ and as a result $(T+(-1)^{n+1})\Delta_n=\Delta_{n+1}\partial+(-1)^n\partial\Delta_{n+1}$. As I understand it $(T+1)\Delta_1$ is a chain map from $C_*(X)$ to $C_*(X)\otimes C_*(X)[1]$. (If $C_*$ is a chain complex then define the shifted chain complex $C_*[p]$ by $(C_*[p])_n=C_{n+p}$ and $d_n^{C_*[p]}=(-1)^pd_{n+p}^{C_*}$). Hence the differentials of $C_*(X)\otimes C_*(X)[1]$ are $-\partial^{\otimes}$ so using the definition of chain homotopy it should be $(T+1)\Delta_1=\Delta_2\partial-\partial^{\otimes}\Delta_2$.
Do I miss something or does there exist a typo in the book?