Strom defines CW-complexes in his book Modern Classical Homotopy Theory (p. 47, ch. 3.2.1) via composition of pushouts, i.e. given a discrete topological space $X_0$, he constructs $X_{n+1}$ from $X_n$ via a pushout $$ \require{AMScd} \begin{CD} \coprod S^n@>{\alpha_n}>>X_n\\ @V{\coprod i}VV @V{j_n}VV\\ \coprod D^{n+1}@>>>X_{n+1} \end{CD}\text{.} $$ A CW-Complex $X$ is then obtained by taking the colimit of the diagram $$X_0\xrightarrow{j_0} X_1\xrightarrow{j_1}\cdots\xrightarrow{j_{n-1}} X_n\xrightarrow{j_n} X_{n+1}\xrightarrow{j_{n+1}}\cdots$$ He then states that the characteristic map $\chi\colon D^n\to X$ of an $n$-cell $D^n$ is given by the diagram $$ \require{AMScd} \begin{CD} \coprod S^n@>{\alpha_n}>>X_n\\ @V{\coprod i}VV @V{j_n}VV\\ \coprod D^{n+1}@>>>X_{n+1}\\ @A{p}AA @VVV\\ D^n@>{\chi}>>X \end{CD} $$ without giving any explanation on how to understand $p$. The only natural definition for $p$ I can think of is that it is the composition of the quotient map $$D^n\to D^n/\partial D^n\cong S^n$$ and an inclusion into the coproduct, such that $p(D^n)=i(S^n)$. But even then this doesn't make sense to me, assume we have $X=S^2$, constructed from one $0$-cell and one $2$-cell. To find $\chi\colon D^2\to X$, we would need a map $p\colon D^2\to\coprod D^3$. But by construction, $\coprod D^3$ should be empty, shouldn't it?
So did I miss something here, or is there an error in Stroms book and it should really be $p\colon D^{n}\to\coprod D^{n}$