What are the auto-equivalences of the category of groups?
Does the category of groups have any nontrivial automorphisms? (an automorphism of a category being a functor from the category to itself with an inverse functor, where the composite both ways is the identity on objects and on hom-sets. By "nontrivial" I mean an automorphism that does not send every object to an isomorphic object).
Probably not, but I don't know how one would prove it. I think I have the beginning of a proof for the category of finite groups, as follows: first, it is clear that a group of prime order must go to a group of prime order, since these are the only groups that have exactly two subgroups (having two subgroups is invariant under category automorphisms). Next, we can use the number of embeddings of the group of order p in the group of order $p^2$ to show that a group of one prime order cannot go to a group of another prime order.
To continue the proof for finite groups or all groups, I suspect we need to use something about the (local) finite presentability of groups, but I am not sure how.
What about other categories, such as abelian groups, commutative unital rings, and modules over a fixed commutative unital ring? The only kinds of nontrivial automorphisms that I know of are the "opposite operation" functor for monoids, noncommutative rings, and similar noncommutative structures.