The title, as you may know already exists somewhere in this wiki page https://en.wikipedia.org/wiki/Singular_homology but without question mark. My question has to do with the proof of the above. I have no idea unfortunately how such a statement can be proved through category theory or if needs a kind of a more intuitive/geometrical approach.
Moreover is that true for singular cohomology as well?
References with a proof or conversation of the above statements are really welcome!