I have encountered with following problem while I was learning homotopy theory.

Let $A\rightarrow X$ and $B\rightarrow Y$ are cofibrations (in a good category). Is it true that a map $A\times Y\cup_{A\times B} X\times B \rightarrow X\times Y$ is a cofibration?

The case $B=S^{n-1}$, $Y=D^n$ was given at some exercises, but I can only prove it by arguing with geometric properties of $B,Y$.

Thank you!