Is the "homotopy category" functor well-defined? $$\def\Cat{\mathbf{Cat}} \def\qCat{\mathbf{qCat}} \def\Catinf{{\mathcal{C}at_\infty}} \def\Catone{{\mathcal{C}at_1}} \def\cC{\mathcal{C}}$$ Let $\Cat$ be the 1-category of small categories, $\qCat$ be the 1-category of small quasi-categories, $\Catinf$ be the $\infty$-category of small $\infty$-categories, and $\Catone$ be the $\infty$-category of small $1$-categories.
A typical construction of the "homotopy category" operation on $\infty$-categories looks starts off like the following construction on $\qCat$:


*

*Take the objects of $\mathrm{h}(X)$ to be $X_0$

*Define the hom-sets to be ...


One then shows that this defines a functor $\qCat \to \Cat$ that preserves equivalences, and thus induces a functor $\Catinf \to \Catone$.
However, this does not obviously induce a $\Cat$-valued functor, since the choice of objects was not made in an equivalence-preserving way. So, I ask

Question: Is there a "homotopy category" functor $\Catinf \to \Cat$?

To better emphasize the problem, consider another approach. the inclusion $\Catone \to \Catinf$ is an accessible, limit preserving functor between presentable $\infty$-categories, and so it has a left adjoint which is the aforementioned functor.
However, $\Cat \to \Catinf$ does not preserve limits, so this method does not work to produce a functor $\Catinf \to \Cat$.
 A: So by this standard, I don't think that the identity functor $Cat_1 \to Cat_1$ satisfies your desired constraints. But perhaps I should elaborate.
What you are describing might be the following. If $W$ is the class of equivalences in $Cat_\infty$, then you have shown that "taking the homotopy category" does not define a functor from $W^{-1} Cat_\infty$, the 1-category obtained by inverting equivalences, to the $1$-category $Cat_1$ of categories and functors: if it did, then it would take equivalences of quasicategories to isomorphisms of categories. The category $Cat_1$ embeds in $Cat_\infty$; a functor becomes an equivalence in $Cat_\infty$ if and only if it was an equivalence in $Cat_1$; the "homotopy category" functor is basically the identity on $Cat_1$; and we also have the result that the identity doesn't extend to a functor from the 1-category $V^{-1} Cat_1$, obtained by inverting equivalences, to the 1-category $Cat_1$.
In a sense, though, these 1-category localizations are ignoring higher structure. Namely, $V^{-1} Cat_1$ is the 1-category of categories and isomorphism classes of functors; similarly $W^{-1} Cat_\infty$ is the 1-category of quasicategories and equivalence classes of functors. We do have a functor $W^{-1} Cat_\infty \to V^{-1} Cat_1$: equivalent quasicategories have equivalent homotopy categories.
This is ... possibly? ... fixable. If we have access to some kind of global choice operator for proper classes, then we can construct a functor $sk: Cat_1 \to Cat_1$ that takes a category $C$ and sends it to a skeleton $sk(C) \subset C$, and construct a natural equivalence from $sk$ to the identity functor. (To do this, we need to choose one representative for every isomorphism class of object in every category $C$ simultaneously.) If we do that, then maybe we can compose the "homotopy category" functor with $sk$ and obtain a functor that does factor through $W^{-1} Cat_\infty$, because $sk$ factors through $V^{-1} Cat_1$. But I'm concerned that this is trying to provide a section of $Cat_1 \to V^{-1} Cat_1$ and I don't think that exists.
However, there are reasons why one might object to this. The idea that every category should be replaced by its skeleton, so that equivalent categories are isomorphic, requires hard set-theoretic theorems or is impossible. It is also, in some sense, solving a fake problem. We already have a well-established foundation for how to deal with equivalences of categories. Except in special cases, we are often not as interested in isomorphisms of categories. $Cat_1$, as a $1$-category, doesn't necessarily reflect our interests.
I would instead contend that the "homotopy" category functor $Cat_\infty \to Cat_1$ should at least reflect the 2-categorical nature of $Cat_1$. It takes quasicategories to categories and functors to functors, but it takes natural equivalences of functors to natural isomorphisms of functors. This means that equivalent quasicategories have equivalent homotopy categories, rather than isomorphic ones, and it seems likely that we shouldn't expect more.
