$newcommand{Hom}{mathrm{Hom}}newcommand{Set}{mathrm{Set}}newcommand{hom}{mathrm{hom}}$In this example one cannot really say that the internal Hom is “exotic”, but at least it is morally different from the external Hom.

Let $G$ be a group and let $Set(G)$ be the category of sets with a $G$-action. Morphisms of this category are mappings of sets which commute with the action. This category has finite products, given by the products of the underlying sets. In fact, it is cartesian closed: The internal Hom $hom(X,Y)$ is the set of *all* mappings $X to Y$, with the $G$-action given by conjugation: $$sigma cdot f := X xrightarrow{cdot sigma^{-1}} X xrightarrow{f} Y xrightarrow{cdot sigma} Y. $$

Note however that in a monoidally closed category, the external Hom can always be recoverd from the internal Hom as the set of its global elements:

$$ Hom(X,Y) cong Gamma(hom(X,Y)) := Hom( I, hom(X,Y)).$$

Here, $I$ is the unit object of the monoidal category.

In the above example, global elements of some $G$-set $Z$ are morphisms from the one-point $G$-set to $Z$. These correspond precisely to the fixed points of $Z$ under the $G$-action: $Gamma(Z) = Z^G$. In particular, the global elements of the internal Hom are mappings invariant under conjugation – i.e. morphisms of $G$-sets.