The simpler question is to study the 2-groupoid $\mathrm{Aut}(\mathsf{Cat})$ of all autoequivalences of the category of (small) categories (i.e. the groupoid of equivalences of categories $\mathsf{Cat} \to \mathsf{Cat}$). This is equivalent to $\mathbb{Z}/2$ considered as a locally discrete 2-groupoid: the terminal object $\mathbb{1}$ is fixed up to unique isomorphism, and the arrow category $\mathbb{2}$ is the unique minimal generator (i.e. it is a generator and no proper subobject is a generator) so also fixed up to isomorphism. Since every category is functorially a colimit of copies of $\mathbb{2}$, once we decide whether our autoequivalence fixes or swaps the two maps $\mathbb{1} \to \mathbb{2}$ we've determined the autoequivalence up to a canonical isomorphism. And of course the nontrivial autoequivalence is the "opposite category" functor $\mathcal{C} \mapsto \mathcal{C}^\mathrm{op}$.


This is a somewhat artificial question, since an autoequivalence of the category $\mathsf{Cat}$ must respect the notion of isomorphism of categories, whereas all we should really want to respect is equivalence of categories. One way to fix this would be to consider $\mathsf{Cat}$ as a 2-category, and ask about the 3-groupoid of autobiequivalences of $\mathsf{Cat}$. But this starts to involve heavy machinery, and the problem will only get worse if we want to go up the categorical ladder. Hence a more "homotopical" approach:


**Question:** What is the biequivalence type of the 2-groupoid $\mathrm{Aut}(\mathrm{ho}\mathsf{Cat})$ of autoequivalences of $\mathrm{ho}\mathsf{Cat}$, the category of small categories localized at the equivalences of categories? Equivalently, $\mathrm{ho}\mathsf{Cat}$ is the category of small categories modulo the congruence identifying naturally isomorphic functors. In particular, is $\mathcal{C} \mapsto \mathcal{C}^\mathrm{op}$ isomorphic to the identity (presumably the answer is no)? Are there any other autoequivalences, essentially (probably not, but you never know)?


A variant would be to ask about the 2-groupoid of functors $\mathsf{Cat} \to \mathsf{Cat}$ which preserve equivalences of categories and induce equivalences on $\mathrm{ho}\mathsf{Cat}$.


I'm aware that Toën [showed](http://arxiv.org/abs/math/0409598) in some sense that the $(\infty,1)$-category of $(\infty,1)$ categories has an automorphism group of $\mathbb{Z}/2$, and Barwick and Schommer-Pries [generalized this](http://arxiv.org/abs/1112.0040) to $(\infty,n)$-categories for larger $n$. This question is meant to be an easier / more elementary warmup to those questions.


The approach that worked for $\mathsf{Cat}$ certainly isn't going to work for $\mathrm{Ho}\mathsf{Cat}$: Freyd showed ("On the concreteness of certain categories." Symposia Mathematica. Vol. 4. 1968-9, pp. 431-56) that $\mathrm{Ho}\mathsf{Cat}$ is not concrete, so it certainly can't have a generating object, never mind a colimit-dense object.


By playing around, I've convinced myself at least that the indiscrete categories (the equivalence class of the terminal category) can be characterized in $\mathrm{Ho}\mathsf{Cat}$ by the fact that all functors into it are isomorphic (consider functors from the walking equalizer diagram to see that such a category is a preorder and functors from the terminal category to see that all objects are isomorphic). So $\mathrm{Ho}\mathsf{Cat}(\mathbb{1},-)$ tells us how many isomorphism classes of objects a category has, and what isomorphism classes of functors are doing to them. But that's about all I've got.



Does the category of categories-mod-natural-isomorphism have any nonobvious autoequivalences?
$\newcommand{\hoCat}{\mathrm{ho}\mathsf{Cat}}
\newcommand{\Cat}{\mathsf{Cat}}
\newcommand{\Gaunt}{\mathsf{Gaunt}}
\newcommand{\hoGaunt}{\mathrm{ho}\mathsf{Gaunt}}
\newcommand{\Free}{\mathsf{Free}}
\newcommand{\hoFree}{\mathrm{ho}\mathsf{Free}}
\newcommand{\comp}{\mathrm{comp}}
\newcommand{\inv}{^{-1}}
\require{AMScd}$


Chris Schommer-Pries' argument in the comments, can be beefed up to show that indeed, essentially the only autoequivalences of $\hoCat$ are the identity and taking the opposite category.


**Summary.**
Use Chris's argument to show that an autoequivalence $F: \hoCat \to \hoCat$ restricts to the identity on the free categories $\Free = \hoFree$ (possibly after composing with the known autoequivalence $(-)^\mathrm{op}: \hoCat \to \hoCat$). As Chris then argues, $\mathsf{Free}$ forms a (large) [regular generator](http://ncatlab.org/nlab/show/separator#strengthened_separators) in $\hoCat$, and it follows that $F$ is essentially the identity on objects. We just need the slightly stronger statement that $\Free$ is a (large) [dense generator](http://ncatlab.org/nlab/show/dense+subcategory) to show that $F$ is also the identity on morphisms. It turns out that the $\Cat$-functoriality of the presentations of categories is enough to guarantee this, even though these presentations are not actually functorial on $\hoCat$. In fact, this argument generalizes to show that if the category of algebras for a monad is modded out by a congruence relation, then the free algebras still form a dense generator in the resulting category $\widetilde{\mathcal{C}^T}$ *if* the canonical coequalizer diagrams descend.


**Details.**


* As I said in the question statement, the terminal category $\mathrm{pt}$ remains terminal in $\hoCat$, so it is preserved by any autoequivalence. My argument in the question statement is overly complicated: it's clear that when modding out by a congruence, the terminal object remains terminal.
* As Chris argues, the subcategory $\mathrm{ho}\mathsf{Pre}$ of preorders is preserved by any autoequivalence because it is the class of categories $P$ with $\hoCat(D,P) \to \mathsf{Set}(\hoCat(\mathrm{pt},D),\hoCat(\mathrm{pt},P))$ injective for every category $D$. One direction is clear, the other comes from taking $D$ to be the walking equalizer diagram. Now, since $\mathrm{ho}\mathsf{Pre} = \mathrm{ho}\mathsf{Pos} = \mathsf{Pos}$, our autoequivalence of $\hoCat$ restricts to either the identity or opposite on $\mathsf{Pos}$ by the same argument I gave in the question statement characterizing the autoequivalences of $\Cat$. After composing with the "opposite" autoequivalence of $\hoCat$, we can assume our autoequivalence is the identity on $\mathsf{Pos}$.
* As Chris argues, in particular the walking arrow $\mathbb{2}$ is fixed. So the groupoids are also fixed: they are characterized as the categories $G$ such that every functor $\mathbb{2} \to G$ extends up to natural isomorphism along the functor $\mathbb{2} \to \mathrm{pt}$. The connected groupoids are also fixed because $\hoCat(\mathrm{pt},-)$ tells us how many isomorphism classes a category has. Note that we don't know the autoequivalence fixes each individual groupoid, but only that the class of groupoids is preserved, but this will be good enough for the next step.
* As Chris argues, the [gaunt categories](http://ncatlab.org/nlab/show/gaunt+category) $\hoGaunt$ are fixed because they are the categories such that every functor from a connected groupoid extends along the functor to $\mathrm{pt}$. We have $\Gaunt = \hoGaunt$, and again the argument given in the question statement for $\mathsf{Cat}$ shows that the autoequivalence restricts to the identity on $\Gaunt$. In particular, it's the identity on the subcategory $\Free$ of free categories.
* As Chris argues, the $\Cat$ - coequalizer $P^2X \stackrel{\to}{\to} PX \overset{\comp\_X}{\to} X$ descends to $\hoCat$ (in particular, $\Free$ is a regular generator in $\hoCat$). Here $P$ is the "path" comonad on $\Cat$, induced by the forgetful functor to graphs, and the coequalizer is the canonical one: one leg $\mu\_X$ de-parenthesizes a path of paths, while the other $P\comp\_X$ composes each individual path in a path of paths; the coequalizer $\comp\_X$ composes a path in $X$. The reason the coequalizer descends is that if $F: PX \to Y$ coequalizes $\mu\_X$ and $P\comp\_X$ up to a natural isomorphism $\alpha$, then for every path $\gamma\_1, \dots, \gamma\_k$ in $X$ we have $F(\gamma\_k \circ \cdots \circ \gamma\_1) = \alpha \circ F(\gamma\_k) \circ \cdots \circ F(\gamma\_1) \circ \alpha\inv$. But in the unary case, this tells us that $F(\gamma\_i) = \alpha \circ F(\gamma\_i) \circ \alpha\inv$. So we can commute the $\alpha$ down to the end to get $F(\gamma\_k \circ \cdots \circ \gamma\_1) = F(\gamma\_k) \circ \cdots \circ F(\gamma\_1) \circ \alpha \circ \alpha\inv = F(\gamma\_k) \circ \cdots \circ F(\gamma\_1)$. That is $F$ already coequalizes the fork strictly, so it descends to a functor $X \to Y$. Uniqueness of the factorization in $\hoCat$ is easy to see.
* Let $i\_\Free: \Free \to \hoCat$ be the inclusion functor. Let $\alpha: \hoCat(i\_\Free - , X) \implies \hoCat(i\_\Free - , Y)$ be a natural transformation. To show that $\Free$ is dense in $\hoCat$, we just need to show that $\alpha$ is induced by some $\bar\alpha: X \to Y$ (we know that such a $\bar\alpha$ is unique because we've shown that $\Free$ is a regular generator, and hence a generator, in $\hoCat$). There is an obvious candidate: by naturality $\alpha\_{PX}(\comp): PX \to Y$ coequalizes the two functors $P^2X \to PX$ up to natural isomorphism, so by the previous part it descends to a functor $\bar\alpha: X \to Y$. So much is true of any regular generator; the trick is to see that $\bar\alpha$ really induces the natural transformation $\alpha$, i.e. that $\bar\alpha \circ f \cong \alpha\_A(f)$ for any $f: A \to X$ with $A \in \Free$, leading to the following diagram:
$$
\begin{CD}
PA @>\comp\_A>> A;\\
@VPfVV @VfVV \searrow^{\alpha\_A(f)} \\
PX @>\comp\_X>> X @>\bar\alpha>> Y;
\end{CD}
$$
Since the square commutes and the outside commutes up to natural isomorphism and $\comp\_A$ is epi in $\hoCat$, the triangle also commutes up to natural isomorphism as desired.


The principle used to analyze the auotequivalences of $\Cat$ in the question statement, and then $\mathsf{Pos}$, and then $\Gaunt$, and finally $\hoCat$, is that an equivalence (or any colimit-preserving functor) is determined up to natural isomorphism by its action on a dense generator. One way to see this is that a dense generator $\mathcal{A} \subseteq \mathcal{C}$ embeds $\mathcal{C}$ as a full subcategory of $\mathrm{Fun}(\mathcal{A}^\mathrm{op},\mathsf{Set})$ containing $\mathcal{A}$ as the representables. Each object is (canonically) a colimit of representables, and each morphism is (canonically) a colimit of maps between representables, so the value of the functor on them is determined by colimit preservation and where it sends the representables.


