Unique (Relation)

Definition

Let $R:X\to Y\to \mathrm{Prop}$ be a relation.
$R$ is considered left-unique or unique in $X$ iff for every $x$ in $X$, there is at most one $y$ s.t. $Rxy$.
Likewise $R$ is right-unique or unique in $Y$ iff for every $y$ in $Y$, there is at most one $x$ s.t. $Rxy$.

This analogy extends to other relation arities.