I think n.piskunov is on the right track.
I would use it as the mathematical analogy found here: https://math.stackexchange.com/questions/247380/analogy-between-one-and-only-one-and-if-and-only-if
"existence of at most one such X that satisfies Y."
As an example, consider these relations:
A student, Alice, can only have one dorm room at at time. A dorm room can only house one student at a time (for the sake of this example). Next year, Alice will be assigned a new dorm room, and at that point her dorm room from this year will be assigned to a new student.
Alice can have one and only one login (e.g. a11235) and that login can only be assigned to Alice. When Alice graduates, no one else can be assigned the login a11235.