If is a monoidal category, the unit -category is the -enriched category having one object, say , with , the monoidal unit object of , and the composition and identity-assigning morphisms being the canonical coherence isomorphism and the identity arrow , respectively.
From a -categorical point of view, the unit enriched category satisfies the property that (the category of -endomorphisms of ) is a category with one object, and the -functor is the functor returning the underlying category of an enriched category.
In other words, the -category plays a role in enriched category theory similar to the role played by the terminal category in ordinary unenriched category theory. (In fact, the terminal category is the unit Set-category.) For instance, objects of a -category can be identified with -functors .
Note that in general the unit -category is not a terminal object of , just as is not usually terminal in . We can see in particular that when is closed and is not terminal, is not terminal since embeds in .