Why the Objects of Homotopy Category not Homotopy Classes of Spaces?

Question

A homotopy category is a category whose objects are topological spaces and whose morphisms are homotopy classes of continuous functions. I wonder why the objects are spaces, instead of homotopy classes of spaces? Does the category whose objects are homotopy classes of spaces and whose morphisms are homotopy classes of continuous functions make sense? If we compare these two definitions, what are their differences and similarities?