In a cartesian closed category, the exponential object [A,B] basically internalizes the collection of morphisms from A to B.
Is there some similar notion that internalizes the isomorphisms between A and B? What about the unique isomorphisms?
It seems like you could get somewhere with subobject classifiers, but maybe there is something more elementary?
My apologies if this is well-known; I haven't been able to dig anything up via searching.