Let $Y,X$ be simplical CW-complexes (By that I mean functors $\Delta^{op}\to CW$. In other words, for each natural number $n$ there exist CW-complexes $X_n$ and $Y_n$ and there are face and degeneracy maps, satifying the usual simplical identities.)
Further let $f:X\to Y$ be a simplicial map, such that on every level (simplicially, not on skeletons) $f_n:X_n\to Y_n$ is a cellular inclusion onto some connected components of $Y$. In other words, $X_n$ is just the collection of some components of $Y_n$.
Now assume that $Y$ is contractible (after geometrc realisation). Does this imply that $X$ is also contractible?
On each level we have (non-canonical) retractions $r_n:Y_n\to X_n$ such that $r_n\circ f_n=id_X$, but I don't see if one can choose them compatibly to obtain a retraction $r:Y\to X$.
(In my situation both $X$ and $Y$ are in fact classifying spaces of sufficiently horrible simplicial comma-categories. Whereas I believe that what I want is true, I am not sure if this topological datum is actually enough. But it seems reasonable.)