homology - disagreement between two definitions of the singular boundary map

The point Rotman is trying to make is the following: if you have a singular $q$-simplex $sigma:Delta^qto X$, then for example the restriction $sigma|_{[e_0,e_2,dots,e_q]}$ is not a singular $(q-1)$-simplex, simply because its domain $[e_0,e_2,dots,e_q]$ is not the standard simplex $Delta^{q-1}$, which is instead $[e_0,e_1,dots,e_{q-1}]$.

He fixes this by composing with the face maps $varepsilon$, so as to get the domains right.