I want to show that given a subcomplex $A$ of a CW-complex say $X$ and a cellular map $f\colon A \to Y$ where $Y$ is another CW complex, the pushout $X\cup_A Y$ inherits CW-structure. I have an idea on how I would go about this but I am finding it difficult to put them together.
First of all, its natural to let the $n$-th skleton to be $X_n \cup_{A_n} Y_n$ and that a I believe it holds that $$ X \cup_A Y = \bigcup_{n\geq-1} X_n \cup_{A_n} Y_n. $$ The reason I believe this holds is that think pushout is preserved under taking unions assuming compatibility(if they agree on the intersections?) of the pushouts. In the case of disjoint unions, I think this is immediate. I would like to know if this is actually the reason.
Secondly I want to show that $X_n\cup_{A_n}Y_n$ is obtained from $X_{n-1}\cup_{A_{n-1}}Y_{n-1}$ by attaching cells. If my intuitions are correct, this consists of $n$-cells in $Y_n$ and the $n$-cells in $X_n$ that are not in $A$. Therefore, I want to construct a pushout square $$\require{AMScd} \begin{CD} I_n\times \partial D^n \amalg J_n\times\partial D^n @>{q\circ x\amalg y}>> X_{n-1}\cup_{A_{n-1}}Y_{n-1}\\ @VVV @VVV \\ I_n\times D^n \amalg J_n\times D^n @>{}>> X_{n}\cup_{A_{n}}Y_{n} \end{CD}$$ where $q\circ x \amalg y$ is first the component-wise attaching map for $X_n\setminus A$ and $Y$ and then the restriction of the quotient to the pushout $X_n\cup_{A_{n}} Y_n$. But I have no idea how to show that this is a pushout square. Any help is appreciated