Homotopy equivalence between two mapping tori of compositions
For any maps there is defined a homotopy equivalence
Here, T(f) denotes the mapping torus of a self-map (not necessarily a homeomorphism). It is very surprising to me that this holds with no extra conditions on d and s. I'm guessing that the homotopy inverse is the map:
If the above is a genuine homotopy inverse, then the map:
would have to be homotopic to the identity somehow. However, after banging my head against the wall on this for a while I can't come up with a valid homotopy. So my questions are:
Is the map I've defined above actually a homotopy inverse? If so, what is the homotopy from the composition I wrote down above to the identity map?
Is there a better one that makes the homotopy obvious?