Suppose that $X$, $Y$ and $Z$ are topological spaces, with $A\subset X$, a map $f:A\rightarrow Y$, and a homotopy equivalence $\phi:Y\rightarrow Z$. It seems fair to think that the adjunction spaces $Y\cup_{f}X$ and $Z\cup_{\phi\circ f}X$ will be homotopy equivalent provided that $(X,A)$ has the homotopy extension property.
A reasonable candidate for a homotopy equivalence seems to arise from the map $(Id,\phi):X+Y\rightarrow X+Z$ after passing to the quotient ($X+Y$ denotes disjoint union, and $(Id,\phi)$ is the map defined to be the identity on $X$ and $\phi$ on Y).
Any suggestions will be appreciated.
Thanks!
