I am reading John W. Milnor, Morse Theory, though it might be a bit difficult, and I was showing in Lemma 3.6 that $k\circ l$ is homotopic to identity map. (Until here is for those who have the specified book.) There, I found I have to construct a homotopy $(D^{\lambda}\cup_{\varphi}X)\times [0, 1]\to D^{\lambda}\cup_{\varphi}X$.
In order to define this continuous map, it seems to me that if homeomorphism like $(X\cup_{\varphi}Y)\times Z\approx (X\times Z)\cup_{\varphi\times id_Z}(Y\times Z) $ holds, then it is easier to define the map because I can make a better use of the universal mapping property. And, this is apparently true according to my intuition. Therefore, I would like to ask if that holds. Any comments appreciated.