Rel, bicategory of relations, allegory
left and right euclidean;
extensional, well-founded relations.
A binary relation from a set $X$ to a set $Y$ is called entire if every element of $X$ is related to at least one element of $Y$. This includes most examples of what the pre-Bourbaki literature calls a (total) multi-valued function (although that term usually implied some continuity or analyticity properties as well). An entire relation is sometimes called total, although that has another meaning in the theory of partial orders; see total relation.
A function is precisely a relation that is both entire and functional.
Like any relation, an entire relation $r$ can be viewed as a span
Such a span is a relation iff the pairing map from the graph $\Gamma_r$ to $X \times Y$ is an injection, and such a relation is entire iff the projection map $\pi_r$ is a surjection.
The axiom of choice says precisely that every entire relation contains a function. Failing that, the COSHEP axiom may be interpreted to say that, given $X$, there is a single surjection $\pi_X: \Gamma_X \to X$ such that every entire relation from $X$ contains a relation given by a span whose left leg is $\pi_X$. In any case, entire relations may be preferable to functions in some contexts where the axiom of choice fails.
When internalising entire relations to a site, one may want to replace the projection map $\pi_r: \Gamma_r \to X$ by a covering family.