Consider the following interesting theorem (7.5.7, p.294 in *Topology and Groupoids* by Ronald Brown):

**Gluing theorem for adjunction spaces:**
*Suppose that we have the following commutative diagram of topological spaces and continuous maps:*

    ![alt text][1]

*where* $\varphi_{A}$, $\varphi_{X}$ *and* $\varphi_{Y}$ *are homotopy equivalences, and the inclusions* $i$ *and* $i'$ *are closed cofibrations. Then the map*

$$
\varphi:X\cup_{f}Y\rightarrow X'\cup_{f\phantom{l}'}Y'
$$

*induced by* $\varphi_{A}$, $\varphi_{X}$ *and* $\varphi_{Y}$ *is a homotopy equivalence*. 

[In this post][2] I asked a question that would become a corollary if the gluing theorem were true for arbitrary cofibrations $i$ and $i'$. **I would like to know if this is the case. Otherwise, does anybody know of a counterexample?**

Perhaps, it is possible that although the map $\varphi$ may not in general be a homotopy equivalence when the cofibrations $i$ and $i'$ are arbitrary, the adjunction spaces $Y\cup_{f}X$ and $Y'\cup_{f'}X'$ will nonetheless be homotopy equivalent. A possible approach to this would be to try to find a third space containing both as deformation retracts. 


**Note on notation:** That an inclusion map $j:B\rightarrow Z$ is a closed cofibration means that it is a cofibration (i.e., the pair $(B,Z)$ has the homotopy extension property) and the set $B$ is closed in $X'$.
 
Thank you


  [1]: https://i.sstatic.net/wwM62.jpg
  [2]: https://mathoverflow.net/questions/94830