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.