Let $A$ be a nonempty topological space, and suppose $X,Z$ are topological spaces that contain $A$ as a subspace. Thus we have two pairs $(X,A)$ and $(Z,A)$. Then let $f:Z\to X$ be a continuous map that restricts to the identity on $A$ ($f|_A=\text{id}_A$). Let $M=M_f=(Z\times I \coprod X )/(z,1)\sim f(z)$ be the mapping cylinder of $f$, and let $W$ denote the quotient space obtained from $M$ by collapsing each segment $a \times I$ to a point for $a \in A$.
In Hatcher's Algebraic Topology, Hatcher calls the space $W$ as the relative mapping cylinder of $f$ and uses it in the proof of Propositions 4.15 and 4.18, using the following properties without proof:
$W$ contains $X$ and $Z$ as subspaces.
$W$ deformation retracts onto $X$.
$W$ is a CW complex containing both $X$ and $Z$ as subcomplexes if $(X,A)$ and $(Z,A)$ are CW pairs and $f$ is a cellular map.
How can I see that these are indeed true?
I know that $M$ contains $X$ and $Z$ as subspaces and that $M$ deformation retracts onto $X$. If 1 is true, I can then see that the deformation retraction of $M$ onto $X$ induces a deformation retract of $W$ onto $X$ so 2 will follow.