TITLE: Categories whose auto-equivalences are naturally isomorphic to the identity
QUESTION [5 upvotes]: Are there any useful characterisation of categories whose auto-equivalences are all naturally isomorphic to the identity? For example, I read in this thread, What are the auto-equivalences of the category of groups? , that the category of groups is one such category. Is there some nice general property that I can use to check whether or not a category admits of auto-equivalences that are not naturally isomorphic to the identity? If not, it'd be useful just to find some more interesting examples of categories with this property. 
More specifically, what examples (if any) are there of auto-equivalences that send every object to an isomorphic object but are not naturally isomorphic to the identity? For instance, I know that any auto-equivalence of SETS must send every object to an isomorphic object, but is it true that any such equivalence is naturally isomorphic to the identity? If so, is this a general property of toposes or just a special case?

REPLY [4 votes]: Since you mention toposes, it may be worth pointing out a simple example of a topos that admits a nonidentity automorphism: $\mathrm{Set}\times\mathrm{Set}$ has an automorphism that swaps the two copies.
On the other hand, the proof that every automorphism of $\mathrm{Set}$ is the identity can be internalized to show that every indexed automorphism of any topos $\mathcal{E}$ is the identity.  Here an "indexed automorphism" consists of automorphisms $\mathcal{E}/X \simeq \mathcal{E}/X$ for all $X\in\mathcal{E}$ that commute appropriately with the pullback functors $f^*:\mathcal{E}/Y \to \mathcal{E}/X$.  Informally, since every topos "thinks that it is the category of sets", it also "thinks that every automorphism of itself is the identity".  The argument is basically the same: in the world of $\mathcal{E}$-indexed categories, $\mathcal{E}$ itself is the free cocompletion of 1.
Similarly, if $\mathcal{V}$ is a complete and cocomplete closed symmetric monoidal category, then every $\mathcal{V}$-enriched automorphism of $\mathcal{V}$ is the identity.  The proof is again the same: in the world of $\mathcal{V}$-enriched categories, $\mathcal{V}$ itself is the free cocompletion of 1.