Notation of the Isom object I was wondering the follows. Let $\mathcal{C}$ be a Cartesian closed category. Then for each $X,Y\in\mathcal{C}$, we have the exponential objects $X^Y$ and $Y^X$. These are abstract analogues of Hom-set $Map(X,Y)$ if $X$ and $Y$ are sets, and $\mathcal{Hom}(X,Y)$ if $X$ and $Y$ are (pre-)sheaves on some site.
I was wondering if there are also analogues of the objects $Bij(X,Y)$ in case $X$ and $Y$ are sets, and $\mathcal{Iso}(X,Y)$ if $X$ and $Y$ are sheaves on a site. I am specifically wondering what they are called, and how they are denoted. 
EDIT: It is easy to see that the Iso object (what I denote by $X^{Y*}$, in the same vein as unit groups of rings) exists if the category is closed under finite limits, and if $X^Y$ and $Y^X$ both exist. I just want to know the official notation and name.

If you have equalizers (and thus pullbacks too) you can define the subobject
$$ \text{InversePairs}(X,Y) = \{ (g,f) \in X^Y \times Y^X \mid f \circ g = 1_X \wedge g \circ f = 1_Y \} $$
I haven't checked, but the composing with the projection
$$ \text{InversePairs}(X,Y) \to X^Y \times Y^X \to Y^X $$
should give a monic map: e.g. the pullback of this arrow by itself should be isomorphic to
$$ \{ (g,g,f) \subseteq X^Y \times X^Y \times Y^X \mid g \circ f = 1_X \wedge f \circ g = 1_Y \} $$
(with the usual projections) by applying the usual argument showing inverses are unique, and this isomorphic to $\text{InversePairs}(X,Y)$.
Thus, the subobject $\text{InversePairs}(X,Y) \to Y^X $ is the subobject of isomorphisms.
