In the same way that internal homs in a cartesian monoidal category behave as an internal version of hom-sets between two objects, the object of isomorphisms should behave as an internal version of the sub-hom-set of isomorphisms between two objects.

$X \mapsto \mathrm{Hom}_{\mathrm{Core}(C/X)}(A \times X, B \times X)$

which takes an object $X \in C$ to a morphism between objects $A \times X$ and $B \times X$ in the core of the slice category$C/X$. A representing object of the above functor, if it exists, is the object of isomorphisms between $A$ and $B$ and is denoted as $[A, B]_\mathrm{iso}$.

Properties

If the category is a cartesian closed category, then there is a monomorphism$i:\mathrm{Mono}_C([A, B]_\mathrm{iso}, [A, B])$ between the object of isomorphisms between $A$ and $B$ and the internal hom from $A$ to $B$.