The question is pretty much the same as >here<. This wikipedia article claims that in "Homotopy category of CW complexes" there is an isomorphism which is not bijective in the underlying sets, while this category should admit underlying set (e.g. faithful functor to Set).
I am not really familiar with CW complexes or Homotopy theory really. Can someone elaborate on what is going on? In the linked post, there is a proof that whenever $f$ is iso in cat. $\mathcal{C}$, then its image $F(f)$ under any functor $F$ must be iso, because $F(f^{-1})$ is its inverse.
The condition for an arrow $g:y\to x$ in a category $\mathbf C$ to be the inverse of $f:x\to y$ is equational: $fg=\mathrm{id}_y$ and $gf=\mathrm{id}_x$. Hence it will be preserved by any functor, by which I mean that $F(g)$ is a inverse for $F(f)$ for any functor $F$ with domain $\mathbf C$: indeed, $F(g)F(f) = F(gf) = F(\mathrm{id}_x) = \mathrm{id}_{F(x)}$ and similarly for the other equation.
Hence, if you have any functor from $\mathbf C \to \mathsf{Set}$, it will map isomorphism to bijections. In the Wikipedia article, it is just said "every object admits an underlying set", with no indication that this mapping should be a functor (or even extended to maps): indeed, in the category $\mathsf{Hot}$, the objects are CW-complexes and the morphisms are the continuous functions modulo homotopy between them, so you can define a mapping $\mathsf{Hot}\rightsquigarrow \mathsf{Set}$ on objects by mapping a CW-complex to its underlying set (just forget the topology), but in no way can you complete that into a functor.