I have a question.
Given a nulhomotopic map $f : X \rightarrow Y$, we can define a prism operator $P : C_n(X) \rightarrow C_{n+1}(Y)$ between singular chains.
Then do we have a cellular prism operator $Q : C_n^{CW}(X) \rightarrow C_{n+1}^{CW}(Y)$, where $C_{*}^{CW}$ denotes cellular chains?
If not, is there some kind of condition where we can get such $Q$? I think we can kind of extend the prism operator $P$ using acyclic models, but I do not see how (or maybe there's another way of doing it?).
PS) Edited to one question only.
Let $h\colon X\times I\to Y$ be a cellular homotopy between cellular maps $f$ and $g$. There is a canonical isomorphism $$C^\mathrm{CW}_\ast(X\times I)\cong C^\mathrm{CW}_\ast(X)\otimes C^\mathrm{CW}_\ast(I)$$ coming from the product CW-structure on $X\times I$. Now $C_\ast^\mathrm{CW}(I)$ has $[0]$ and $[1]$ as basis elements in degree $0$ and $[I]$ in degree $1$. Furthermore, $d([I]) = [0]-[1]$. If we put $s(x) = (-1)^iC_\ast^\mathrm{CW}(h)(x\otimes [I])$ for $x\in C_i^\mathrm{CW}(X)$, then \begin{align}d_Y(s(x)) &= (-1)^id_Y(h_\ast(x\otimes [I]) )\\ &=(-1)^ih_\ast d_{X\times I}(x\otimes [I]) \\ &= (-1)^i h_\ast(d_Xx\otimes [I]+(-1)^ix\otimes d[I]) \\ &= h_\ast(x\otimes d[I])-(-1)^{i-1}h_\ast(d_Xx\otimes [I]) \\ &= h_\ast (x\otimes [0])-h_\ast(x\otimes [1])-s(d_Xx)\\ &=f_\ast(x)-g_\ast(x)-s(d_Xx),\end{align} so this yields a chain homotopy from $C_\ast^\mathrm{CW}(f)$ to $C_\ast^\mathrm{CW}(g)$.