Criterion for a map to be homotopy equivalence in terms of its mapping cylinder

Question

I am trying to prove that a map $f: X \to Y $ is a homotopy equivalence iff $j : X \to Z(f)$ is a deformation retract where $Z(f)$ is the mapping cylinder and $X \to Z(f) \to Y$ is the decomposition $f=qj$ into a homotopy equivalence $q$ and a cofibration $j$. I am able to prove that $f$ has a left homotopy inverse iff $j $ has a retraction. But able to proceed further. Any help would be very much appreciated. P.S. I encountered this problem while learning Algebraic Topology from Tammy Tom Dieck's book. This is problem 5.3.1 on p 113 there.