Is the composition of cellular maps cellular?
Related to this, I have another question. (I apologize to asking very similar question.)
Let ${\sf CWcpx}$ be the category of CW complexes and let ${\sf Top}$ be the category of topological spaces. Take a (natural) functor $i:{\sf CWcpx}\to {\sf Top}$.
My question is "Is the composition $g\circ f$ in the image of $i$ while $f$, $g$ are in the image of $i$?".
More precisely, $X$,$Y$,$Y'$,$Z$ be CW complexes such that $Y$ and $Y'$ are homeomorphic as topological spaces. And let $f:X\to Y$, $g:Y'\to Z$ be cellular maps. My question (reformulate version) is here: "Is there any pair $(X',Z')$ of CW complexes and cellular map $h:X'\to Z'$ such that $X \cong X'$, $Z\cong Z'$ as topological spaces and $h$ is equal to $g \circ f$?".
