I understand that my question is probably elementary to someone well-versed in model categories, but the subject is very deep and I wonder whether there is a much simpler answer.

If you localize a ring, some elements get identified, so I assume the same will happen when we localize a category.

Suppose we start with the category of pointed topological spaces and localize it at homotopy equivalences. I assume such thing exists (defined by a universal property) and I will treat it formally.

I came up with the following argument that if $X$ is contractible, $End(X)$ is just the identity - if we compose two maps $f, g$ them with zero, they are the same, and since zero is invertible in $End(X)$, $f=g$.

However, I'm having problems showing that, in general, if $f, g: X \rightarrow Y$ are homotopic, they are equal in the localization. Is it even true, ie. do we recover the homotopy category of topological spaces in the localization?

If not, then how much can we prove? What changes if I localize at weak equivalences? What happens if we limit ourselves to $CW-complexes$?