Hatcher defines the mapping cylinder $M_f$ of a map $f : Z\to X$ as the quotient space of the disjoint union of $Z\times I$ and $X$ under the identifications $(z, 1) \sim f (z)$.
My question is that if we have a map $X\hookrightarrow M_f$, how can we use the long exact sequence of homotopy groups to show that $\pi_n(M_f, X) = 0$ for all $n$? What would the long exact sequence look like in this case?
Also, how can we see that $Z$ is a deformation retract of $M_f$?
I'm referring to the answer posted on this question: CW approximation is unique - question about proof