Let $\mathcal{M}$ be a model category and let $C:\mathcal{M}\to\mathcal{M}$ be a [very good cylinder object][1]. The natural transformations coming with $C$ are denoted by $\gamma^\epsilon_X:X\to CX$ with $\epsilon=0,1$ and $\sigma_X:CX\to X$. In this setting, a *deformation retract* is a map $i:Z\to Y$ such that there exists a retract $r:Y\to Z$ ($ri=1_Y$) such that the maps $ir$ and $1_Y$ are left homotopic, i.e., there exists a map $h:CY\to Y$ such that $h\gamma^0_Y=ir$ and $h\gamma^1_Y=1_Y$. The deformation retract is *strong* if moreover $h$ can be chosen so that $\sigma_Y C(i) = h C(i)$.

> I can prove that $\gamma^0_X:X\to CX$ is a deformation retract, I
> would like to prove that it is a strong deformation retract.

Proof that it is a deformation retract. Consider the commutative diagram 
$\require{AMScd}$
\begin{CD}
    CX \sqcup CX @>\gamma^0_X\sigma_X \sqcup 1_{CX} >> CX\\
    @V \gamma_{CX} V V= @VV \sigma_X V\\
    CCX @>>\sigma_X \sigma_{CX}> X.
\end{CD}
The left vertical map is a cofibration and the right vertical map is a trivial fibration since the cylinder is very good. Thus, the existence of a lift $h:CCX\to CX$.


  [1]: https://ncatlab.org/nlab/show/cylinder+object