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A relation $\mathcal R \subseteq S \times T$ is a correspondence if and only if $\mathcal R$ is both left-total and right-total.

Also defined as

Some sources, for example 1974: P.M. Cohn: Algebra: Volume 1, define a correspondence as a relation with no restrictions upon it.

That is, a correspondence from $S$ to $T$ is any subset of the Cartesian product $S \times T$.

In order to reduce confusion, it is recommended that this usage of correspondence is not used.