I simply don't understand the proofs used by Hatcher on page 528 in Algebraic topology, that if you have a mapping telescope $T(f_1,f_2, ...)$ then
(1) $ T (f_{1}, f_2, ···) ≃ T (g_1, g_2, ···) \iff f_{i}≃ g_{i}$ for each i.
(2) $T (f_1, f_2, ···) ≃ T (f_2, f_3, ···)$
(3) $T (f_1, f_2, ···) ≃ T (f_2f_1, f_4f_3, ···)$
He claims that (2) is "obvious" and moves on, but I don't see how it's remotely obvious. I also find the proofs for the other 2 a bit hand wavy and unsatisfactory
As Max said for (1) only the reverse implication is true. For (2) remember that the mapping cylinder of $f:X \rightarrow Y$ deformation retracts onto $Y$ (just think of squashing the cylinder into Y). The mapping telescope is homeomorphic to the mapping cylinder of $f_1: X \rightarrow T(f_2,f_3,\dots)$, so (2) follows.
(3) is just what happens if you squash in every other section of the mapping cylinder. I recommend drawing a picture.