From "Homotopy category" > "Concrete categories" https://en.wikipedia.org/wiki/Homotopy_category#Concrete_cat... :
> While the objects of a homotopy category are sets (with additional structure), the morphisms are not actual functions between them, but rather a classes of functions (in the naive homotopy category) or "zigzags" of functions (in the homotopy category). Indeed, Freyd showed that neither the naive homotopy category of pointed spaces nor the homotopy category of pointed spaces is a concrete category. That is, there is no faithful functor from these categories to the category of sets.
Here is the statement:
The meaning of no category is every category.
This was already understood by everybody in the field, no doubt. It's just that somebody has to actually say it to someone else in order for the symmetry to break. The link above has the exact description of this, from Terence Tao.