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. A deformation retraction of $X_1\times I$ onto $X_1\times{0}\cup A\times I$ as in prop 0.16 induces a deformation retraction of $X_0\sqcup_F(X_1\times I)$ onto $X_0\sqcup_f X_1$.
My Question:How do A deformation retraction of $X_1\times I$ onto $X_1\times{0}\cup A\times I$ induce a deformation retraction of $X_0\sqcup_F(X_1\times I)$ onto $X_0\sqcup_f X_1$.What's the definition of the deformation retraction of $X_0\sqcup_F(X_1\times I)$ onto $X_0\sqcup_f X_1$ ? What's more,If $A$ is a deformation retract of $B$ and there's a attaching map $f:A\rightarrow X$ can we get a deformation retract of $X\sqcup_f B$ after attached along $A$ via $f$?
