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Definition

A binary relation from a set $X$ to a set $Y$ is called functional if every element of $X$ is related to at most one element of $Y$. Notice that this is the same thing as a partial function, seen from a different point of view. A (total) function is precisely a relation that is both functional and entire.

Properties

Like any relation, a functional relation $r$ can be viewed as a span

Such a span is a relation iff the pairing map from the domain ${\Delta }_r$ to $X\times Y$ is an injection, and such a relation is functional iff the inclusion map ${\iota }_r$ is also an injection. Such a relation is entire iff the inclusion map ${\iota }_r$ is a surjection.

(Total) functions can be characterized as the left adjoints in the bicategory of relations, in other words relations $r:X\to Y$ in $\mathrm{Rel}$ for which there exists $s:Y\to X$ satisfying

in which case it may be proven that $s=r^{\mathrm{op}}$. A relation is functional if and only if $r\circ r^{\mathrm{op}}\le 1_Y$, and is entire if and only if $1_X\le r^{\mathrm{op}}\circ r$.

Further to this: surjectivity of a function $r:X\to Y$ can be expressed as the condition $1_Y\le r\circ r^{\mathrm{op}}$, and injectivity as $r^{\mathrm{op}}\circ r\le 1_X$.