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.



What does an object of isomorphisms look like?
There is a map $[A, B] \times [B, A] \to [A, A] \times [B, B]$ given by composition in each of the possible directions. There is also a map $1 \to [A, A] \times [B, B]$ which picks out $(\text{id}\_A, \text{id}\_B)$. These fit together into a diagram $[A, B] \times [B, A] \to [A, A] \times [B, B] \leftarrow 1$ and the object of isomorphisms is the limit (so pullback) of this diagram. 


