A closed category basically contains its own hom-objects. Flipping this around, is there a word for categories in which every object is a hom-object for a pair of objects in the category? Or categories in which both properties hold?
I'm interested in thinking more generally about the idea that every group's set of automorphisms form a group (but not every group is the automorphism group of a group), and likewise that every monoid's set of endomorphisms form a monoid (and I think vice-versa).
Given any closed category $C$ with unit object $I$ and any object $X$ of $C$, $[I,X]$ is by definition isomorphic to $X$.