I'm trying to understand the proof of homotopy invariance of induced maps on homology. However, I do not really understand the intuition behind this proof and especially what the prism operators (as defined in Hatcher, for example) exactly represent. Should I try to have a precise, geometric intuition of what is happening, or not? I haven't seen much description of the intuition behind this proof in Hatcher (where usually geometric intuition is emphasised, I think), or in any other text.
Thanks in advance.