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In Jeffrey Strom's Modern Classical Homotopy Theory on page $125$, it is stated that

"Now we come to one of the crucial differences between the pointed and the unpointed categories. The mapping cylinder construction can be carried out for any map $f:X\rightarrow Y$ in the category $\mathcal{T}_*$ , but unless the target $Y$ is well-pointed, the resulting inclusion $\iota:X\hookrightarrow M_f$ will not be a pointed cofibration."

$\mathcal{T}_*$ is the category of pointed spaces (actually of pointed compactly generated weak Hausdorff spaces, but this should have no incidence on what follows). The (pointed) mapping cylinder is the pushout in $\mathcal{T}_*$ of diagram $$ \begin{array}{ccc} X & \hookrightarrow & X \rtimes I \\ f\downarrow ~~& & \downarrow \\ Y & \rightarrow & M_f \end{array} $$ with $I$ the interval $[0,1]$, $X \rtimes I=\frac{X\times I}{*\times I}$ the half smash product of $X$ and $I$, and $X\hookrightarrow X\rtimes I$ is the map $x\mapsto [x,0]$. The inclusion that is supposed to be a cofibration is the inclusion resulting from $\iota:X\hookrightarrow X\rtimes I\rightarrow M_f$ with the first inclusion being $x\mapsto [x,1]$.

Why is it necessary for $Y$ to be well-pointed in order for $\iota$ to be a pointed cofibration? Why can't we construct homotopy extensions the usual way by sliding the pointed mapping cylinder along the homotopy defined on $X$ like a glove (the way we do it for unpointed cofibrations)?

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