Kan extension and singular cohomology

Question

Let $Sm$ denote the category of smooth manifolds and $Top$ the category of topological spaces. Then we have a faithful functor $Sm\rightarrow Top$ which gives a functor $$Ho(Sm)\rightarrow Ho(Top).$$ We know that the homtopy category of topological spaces is equivalent to the homotopy category of CW-complexes. In particular, since CW-complexes are the colimits of spheres, that means that $Ho(Sm)$ is dense in $Ho(Top)$. I'm wondering if we can use this to define the singular cohomology. Assume we have defined the singular cohomology of smooth manifolds. Is there a natural way to extend this functor from $Ho(Sm)^{op}$ to $Ho(Top)^{op}$? The most natural way I know to extend a functor is the Kan extension. Does this do the job? Or is there another way?