I'm reading a proof of the statement that every pair of homotopic maps between topological spaces induces a pair of chain-homotopic chain maps. The document can be found here. In the third page, there is a statement that $$H \circ\left(\sigma_{\left[v_{0}, \ldots, \widehat{v_{j}}, \ldots, v_{n}\right]} \times 1_{I}\right)_{\left[v_{0}, \ldots, v_{i}, w_{i}, \ldots w_{n}\right]}=\begin{aligned}\begin{cases}H_{\left[v_{0}, \ldots, v_{i}, w_{i}, \ldots, \widehat{w_{j}}, \ldots, w_{n}\right]}^{\sigma}&\text{if }i<j\\H_{\left[v_{0}, \ldots, \widehat{v_{j}}, \ldots, v_{i+1}, w_{i+1}, \ldots, w_{n} \right]}^{\sigma}&\text{if }i\geq j\end{cases}\end{aligned}$$
But from my understanding it is supposed to be $$H \circ\left(\sigma_{\left[v_{0}, \ldots, \widehat{v_{j}}, \ldots, v_{n}\right]} \times 1_{I}\right)_{\left[v_{0}, \ldots, v_{i}, w_{i}, \ldots w_{n}\right]}=\begin{aligned}\begin{cases}H_{\left[v_{0}, \ldots, v_{i}, w_{i}, \ldots, \widehat{w_{j-1}}, \ldots, w_{n}\right]}^{\sigma}&\text{if }i<j\\H_{\left[v_{0}, \ldots, \widehat{v_{j}}, \ldots, v_{i}, w_{i}, \ldots, w_{n} \right]}^{\sigma}&\text{if }i\geq j\end{cases}\end{aligned}$$
Why isn't it?
Also, on Hatcher, a similar statement on p. 112:
The expression $H \circ\left(\sigma_{\left[v_{0}, \ldots, \widehat{v_{j}}, \ldots, v_{n}\right]} \times 1_{I}\right)_{\left[v_{0}, \ldots, v_{i}, w_{i}, \ldots w_{n}\right]}$ does not make sense. $\left[v_{0}, \ldots, \widehat{v_{j}}, \ldots, v_{n}\right]$ is an $(n-1)$-simplex and $\left[v_{0}, \ldots, \widehat{v_{j}}, \ldots, v_{n}\right] \times I$ is broken into $n$-simplices. However, $\left[v_{0}, \ldots, v_{i}, w_{i}, \ldots w_{n}\right]$ is an $(n+1)$-simplex. So what should be the restriction?
The $n$-simplices of $\left[v_{0}, \ldots, \widehat{v_{j}}, \ldots, v_{n}\right] \times I$ are $\left[v_{0}, \ldots, v_i, w_i, \ldots, \widehat{w_{j}}, \ldots, w_{n}\right]$ for $i < j$ and $\left[v_{0}, \ldots, \widehat{v_{j}}, \ldots, v_{i+1},w_{i+1}, \ldots, w_{n}\right]$ for $i \ge j$.
This is all you need in the calculation of $P(\partial \sigma)$. See Hatcher's formula. Note that $\Sigma_{i<j}$ is short for $\Sigma_{(i,j) \text{ with } i<j}$