Universal property for chain maps

Question

Given simplicial complexes $X,Y,Z$, such that there exists a chain map $f: C_*(X\times Y)\rightarrow C_*(Z)$, and $\pi_Z$ is the projection $C_*(Z\times Y)\rightarrow C_*(Z)$, does there exist a unique chain map $f':C_*(X\times Y)\rightarrow C_*(Z\times Y)$ such that $\pi_z\circ f'=f$?

Answer 1

No, this is very far from being true. For instance, if $X$, $Y$, and $Z$ are all $0$-dimensional, and $f$ is induced by some map $g:X\times Y\to Z$, this would imply there is at most one map $g':X\times Y\to Z\times Y$ such that $g$ is the first coordinate of $g'$. But this is obviously false if $X$ is nonempty and $Y$ has more than one point, since you could define the second coordinate of $g'$ to be any map $X\times Y\to Y$ at all.