Let $\mathcal{H}$ be the homotopy category of spaces; i.e. $\mathcal{H}$ has as its objects the CW complexes and as its morphisms the homotopy classes of maps between them. I'm trying to understand the slogan that, by Whitehead's theorem, $\mathcal{H}$ is obtained by "formally inverting" the weak homotopy equivalences in $\mathbf{Top}$, but something is not entirely clear to me.
In particular, for any topological spaces $X$, we know there is a CW complex $X'$ and a canonical weak homotopy equivalence $\phi:X'\rightarrow X$, and that $X'$ is unique up to strong homotopy equivalence. The claim I've heard is that this means the object map $X\mapsto X'$ induces a functor $\mathbf{Top}\rightarrow \mathcal{H}$. Can someone enlighten me as to the details of this functor? In particular, I don't see what the map $X'\rightarrow Y'$ in $\mathcal{H}$ induced by an arbitrary map $f:X\rightarrow Y$ in $\mathbf{Top}$ should be; because weak homotopy equivalence is not a symmetric relation there doesn't seem to me to be canonical choice. Sorry if this is a stupid question or if I'm missing something obvious.
Given a map $f:X\to Y$, let $\phi_X:X'\to X$ and $\phi_Y:Y'\to Y$ be the canonical weak equivalences. In particular, since $\phi_Y$ is a weak equivalence, it induces a bijection between homotopy classes of maps $K\to Y$ and homotopy classes of maps $K\to Y'$ for any CW-complex $K$. In particular, taking $K=X'$ and considering the map $f\phi_X:X'\to Y$, this says there is a map $f':X'\to Y'$, unique up to homotopy, such that $\phi_Yf'$ is homotopic to $f\phi_X$. This map $f'$ (or rather, its homotopy class) is where your functor sends $f$. (The fact that this preserves composition follows from the uniqueness of $f'$: if $f:X\to Y$ and $g:Y\to Z$, then $g'f'$ satisfies $\phi_Zg'f'\simeq gf\phi_X$, and so $g'f'$ must be homotopic to $(gf)'$.)