Intuition behind prism operators to prove homotopy invariance of homology

I have not read Hatcher, but Prism's go back to Eilenberg-Steenrod. The idea is simple if $H(x,t)$ is a homotopy on $\Delta_p$ (just assume for the moment it is constant on the boundary), then we can view $H$ as a function on the prism $I\times \Delta_p$. Now the boundary of the prism can be thought of as the difference of the top and the bottom, (the sides are constant). This can be then extended to sums of simplices. And so it shows that homotopy gives a chain-homotopy which induces and isomorphism.

I think the problem arises because $I\times \Delta_p$ is no longer a simplex. Athough personally I have never really understood why this is such a big issue since one can easlily and explicitly write a prism as a sum of simplices.