Mapping cylinder factorization

F2 makes the cylinder CGWH and its free end a closed embedded copy of X. By the defining formulas, $rj(x)=f(x)$. Define $D(y,t)=y$ and $D([x,s],t)=[x,(1-t)s]$. At s=0 both clauses have value f(x); F3 makes the induced homotopy continuous. It starts at the identity, ends at the inclusion followed by r, and fixes Y at every time. This is the claimed strong deformation retraction by F5.

Given initial data $a:M_f\to Z$ and a homotopy $b:X\times I\to Z$ with $b(x,0)=a(j(x))$, prescribe data on the bottom and two vertical sides of the square with coordinates $(s,t)$: $a([x,s])$ on t=0, $b(x,t)$ on s=1, and $a(f(x))$ on s=0. The corner values agree. Keep a(y) constant on Y. Put $\lambda =1\mathrm{/}\max (1-t\mathrm{/}2,2\mathrm{\mid }s-1\mathrm{/}2\mathrm{\mid })$ and $R(s,t)=(1\mathrm{/}2+\lambda (s-1\mathrm{/}2),2+\lambda (t-2))$. The denominator is at least 1/2. If its first term is maximal the second output is zero; otherwise the first output is 0 or 1. Both coordinates lie in I, and on the three designated sides λ=1. Thus R is a continuous retraction onto those sides.

Compose the pasted side data with R. This gives a continuous extension on $X\times I\times I$; its value at s=0 is always a(f(x)). Together with the constant homotopy on Y, F3 descends it to $M_f\times I$. It restricts to a initially and to b on the free end, proving HEP. If a commuting square is $vf=f^{\mathrm{\prime }}u$, the map $[x,s]\mapsto [u(x),s]$, $y\mapsto v(y)$ respects the attaching relation and all displayed height formulas. This proves naturality of the factorization and deformation.

For based data the values on the basepoint track are constant, so the same formulas descend after collapsing that track. Interchange the two vertical sides in step 1.2 to extend a prescribed homotopy at the base s=0 of the reduced cone, keeping the tip s=1 constant. Descent then proves the based cone-base cofibration as well. Empty X gives the unchanged space Y and an empty free-end inclusion.