# Uniqueness of a homotopy category.

For a category with weak equivalences $$(C,W)$$ call $$(ho(C),F)$$ a homotopy category of $$(C,W)$$ where $$ho(C)$$ is a category and $$Q \in Fun^{W}(C,ho(C))$$ is a functor inverting $$W$$ if the following universal property holds:

For any category $$D$$ a functor $$Q$$ induces an equivalence of categories $$Fun(ho(C),D)\rightarrow Fun^W(C,D)$$

My lecture notes claim that $$ho(C)$$ if exists is unique up to unique equivalence of categories by enriched version of Yoneda lemma.

I'm not acquainted with the enriched lemma but I doubt the uniqueness statement because of the following: We can always take $$ho(C)'$$ which differs from $$ho(C)$$ in the way that we take some object $$A\in ho(C)$$ outside the image of $$Q$$, copy-paste it $$5$$ times and call the new category $$ho(C)'$$. Then there is more than one equivalence between $$2$$ homotopy categories considered above.

Am I missing something and if not in which sense the uniqueness should be stated?

P.S. We believe only in locally small categories so formally inverting $$W$$ does not work and the existence of a homotopy category may not always be the case.

P.P.S. Some definitions which I omitted to make it easier(?) to read:

1) $$(C,W)$$ is a category with weak equivalences if $$W\subset Mor(C)$$ s.t. $$2$$ out of $$6$$ holds(i.e. $$fg, gh \in W$$ only if $$f,g,h,fgh \in W$$).

2) $$Fun^{W}(C,ho(C))$$ is a full subcategory of $$Fun(C,ho(C))$$ inverting $$W$$ (i.e. $$f\in W$$ and $$F\in Fun^{W}(C,ho(C))$$ only if $$F(f)$$ is an isomorphism).

Yes, the claim needs to be refined considerably. What is the case is that there is, up to natural isomorphism, a unique equivalence of categories $$ho(C)\to ho(C)'$$ equipped with a natural isomorphism between the composition $$C\to ho(C)\to ho(C)'$$ and the localization functor $$C\to ho(C)'$$. It is straightforward to check this condition from the universal property as given.