Is category equivalence unique up to isomorphism?

Question

Let $\mathbf C$, $\mathbf D$ be categories and $F, F':\mathbf C \to \mathbf D$ and $G, G':\mathbf D \to \mathbf C$ be functors of the shown direction. Is it the case that, if each of $(F,G)$ and $(F',G')$ is an equivalence, then $F\cong F'$ (isomorphic in $\mathbf D^\mathbf C$)?

Initially I thought (without thinking) that there couldn't be more than "one" equivalence (up to isomorphism) between two categories. But, once I tried to prove this, I couldn't find any reason whay it should be so. Was I wrong? Any help will be appreciated.

Answer 1

No; for instance, let $C = D = 2$, the discrete category with two objects, and let $F = G = \mathrm{id}_C$ while $F' = G'$ is the functor exchanging the objects.

Edit to add some more simple examples:

If we think of the poset $(\mathbb Z, <)$ as a category, then the two equivalences $\mathrm{id}_{\mathbb Z}: \mathbb Z \to \mathbb Z$ and $S: \mathbb Z \to \mathbb Z$ are not isomorphic.

If we think of a group as a one-object category (so that functors correspond to group homomorphisms), then for any non-trivial group $G$ the mapping $G \times G \to G \times G$ which swaps the elements is not isomorphic to the identity on $G \times G$. More generally, two group homomorphisms are isomorphic if and only if they are conjugate, so in particular any non-identity automorphism of an abelian group (e.g. negation for most abelian groups) is not isomorphic to the identity on that group.

Answer 2

To get some more intuition on this, note that analogous statements aren't true for other mathematical objects. For example if $G$ and $H$ are isomorphic groups then the set of isomorphisms between them is in bijection with the set of automorphisms of either of them. A group can have more than one distinct automorphism.

Answer 3

Peter Freyd's book "Abelian Categories" contains a discussion of this topic in Exercise B at the end of Chapter 1. He defines the automorphism class group of a category $C$ to be the group of equivalences $C\to C$ modulo those naturally isomorphic to the identity. As an example where this is non-trivial, he gives the category of ordered sets and order-preserving functions. The functor that reverses the ordering is an equivalence but is not isomorphic to the identity.