I am reading a book that has assumed some results along the text that are rather trivial and left our without proofs, so I am looking for a such a proof or a source that goes through the results in more detail.
How can I see that in the $x$-coordinate for the $n$-simplex $\Delta^n= \{(x_1,...,x_n) \in \mathbb{R}^n: 0 \leq x_1 \leq...\leq x_n \leq 1 \}$ the face maps $d_i: \Delta_{n+1}(X) \to \Delta_n(X)$ is of the forms $$d_0(\sigma)(x_1,...,x_n)= \sigma(0,x_1,....,x_n)$$
$$d_i(\sigma)(x_1,...,x_n)=\sigma( x_1,...,x_i,x_i,...,x_n), 0 < i < n+1 $$
$$d_{n+1}(\sigma)(x_1,...,x_n)= \sigma(x_1,...,x_n,1)$$
If we furthermore let $H: X \times I \to Y$ be a homotopy from $f$ to $g$, we may define $h_i: \Delta_n(X) \to \Delta_{n+1}(Y)$ by $$h_i( \sigma)(x_0,...,x_n)= H(\sigma(x_0,...,\hat{x_i},...,x_n),x_i)$$, then the following "trivial" identities are assumed $$d_0h_0=f_{\ast},\;\;\;d_{n+1}h_n=g_{\ast}$$
$$d_ih_j=h_{j-1}d_i, i < j$$
$$d_jh_j=d_jh_{j-1}$$
$$d_ih_j=h_jd_{i-1}, \;\; 1 > j+1$$
How can I see this? For the first result, I believe that a homeomorphism is used $\Delta^n \cong \{ (x_0,...,x_n) \in \mathbb{R}^{n+1}: \sum x_i = 1, x_i \geq 0 \}$