Is there a name for this relation: for all $x$ there is $y$ such that $xRy$, and for all $x,y,z$, if $xRy$ and $xRz$, then $y=z$? Suppose for all $x$ there is $y$ such that $xRy$, and for all $x,y,z$, if $xRy$ and $xRz$, then $y=z$.
Does there exist such a binary relation $R$ on some set such that the above properties are satisfied by $R$?

One set of examples is functions from $A$ to $B$.  For this let $xRy$ mean that $(x,y) \in f$ [using the "function as ordered pairs" formulation]. Then your first requirement expresses that $f$ produces an output $f(x)$ for each $x$ in $A,$ while your second expresses that $f$ is a function.
There may be more  examples.
Edit: In usual math terminology, the term "function" implies it is "single valued". That is, a single input doesn't map to more than one output. That's what your second condition expresses. The first condition really says each element of the domain $A$ maps to at leastone thing in $B$ [the "codomain"].
There is a version of a so-called "partial function" for which not every element of domain needs to map to something in codomain. [I've seen that more used by logicians]
