Question on prop 0.18 by Hatcher

Here is prop 0.18 from Hatcher: if $(X_1,A)$ is CW pair and we have attaching maps $f,g:A\rightarrow X_0$ that are homotopic, then $X_0\sqcup_f X_1$ is homotopy equivalent to $X_0\sqcup_g X_1$ relative to $X_0$.

Hatcher says if $F:A\times I\rightarrow X_0$ is a homotopy from $f$ to $g$, consider the space $X_0\sqcup_F(X_1\times I)$; this contains both $X_0\sqcup_f X_1$ and $X_0\sqcup_g X_1$ as subspaces. In the online errata by Hatcher, he explains that this follows from the fact if $q:X\rightarrow Y$ is any quotient map, $A\subset X$ is a closed saturated set, then $q|_A$ is a quotient map.

My question is but $X_0\sqcup_f X_1$ is not saturated. I think instead Hatcher's observation gives that $X_0\sqcup_F(X_1\times\{0\}\bigcup A\times I)$ is a subspace. Can anyone please explain this to me? Thank you!

Solution 1:

$X_0\sqcup_F(X_1\times\{0\}\bigcup A\times I)$ under quotient topology is homeomorphic to $X_0\sqcup_f X_1$.