Unit in an Allegory

A unit in an Allegory $\mathcal{A}$ is an object $U:\mathcal{A}$ such that $\mathrm{id}:\mathcal{A}(U,U)$ is a Top Element, and every $X:\mathcal{A}$ admits a map $\phi:\mathcal{A}(X,U)$ such that $\mathrm{id}_{X}\leq\phi^{\dagger}\phi$.