Why is an injective relation called left-unique? Given a relation $R \subseteq A \times A$. Then, $R$ might be injective or left-unique. My question is a language question: why is an injective relation called "left-unique"? My feeling would rather to call them (wrongly!) "right-unique", since its "right" elements have a unique (or none) partner on the "left".
I'm looking for a good way of visualizing or explaing the term "left-unique". Thanks!

It makes sense, because an injective relation has the property that $aRb$ and $a'Rb$ forces $a = a'$. That is, you have a uniqueness property on the left side. This is not happening on the right side: uniqueness statements involve equating two things which a priori need not be equal, which is happening on the left; on the right, $b$ is merely a passenger. 
