If $(X_1,A)$ is a CW pair and we have attaching maps $f,g:A\to X_0$ that are homotopic, then $X_0\sqcup_f X_1\simeq X_0\sqcup_g X_1\ \text{rel}\ X_0$.
In the proof of this statement, they claim that if $F:A\times I\to X_0$ is a homotopy from $f$ to $g$, then a deformation retract of $X_1\times I$ onto $X_1\times\{0\}\cup A\times I$ induces a deformation retract of $X_0\sqcup_F(X_1\times I)$ onto $X_0\sqcup_f X_1$. To see this, by the deformation retract of $X_0\sqcup_F(X_1\times I)$ onto $X_0\sqcup_f X_1$, $X_0\sqcup_F(X_1\times I)$ deformation retract onto $X_0\sqcup_F(X_1\times\{0\}\cup A\times I)$. But as $A\times I$ can be identified with the space in $X_0$, $X_0\sqcup_F(X_1\times\{0\}\cup A\times I) = X_0\sqcup_F(X_1\times\{0\})\cong X_0\sqcup_fX_1$.
Is this the correct reason?
