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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).

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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.

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