nLab
unit enriched category

If $V$ is a monoidal category, the unit $V$-category is the $V$-enriched category $ℐ$ having one object, say $\star$, with $ℐ(\star ,\star )=I$, the monoidal unit object of $V$, and the composition and identity-assigning morphisms being the canonical coherence isomorphism $I\otimes I\cong I$ and the identity arrow ${\mathrm{id}}_I$, respectively.

The $V$-category $ℐ$ plays a role in enriched category theory that is 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 $V$-category $𝒜$ can be identified with $V$-functors $ℐ\to 𝒜$. Note that the unit $V$-category is not a terminal object of $V\mathrm{Cat}$, in general, just as $I$ is not usually terminal in $V$.