Characterization of monovalued functions

Question

Let $f$ be a binary relation.

Let $(\bigcap G)\circ f = \bigcap_{g\in G}(g\circ f)$ for every set $G$ of binary relations.

Can we prove that $f$ is monovalued (a function)?

Answer 1

I think not. If $f=\varnothing$ (the empty relation), then $f$ is not a function as long as the underlying set is non-empty, but for any other binary relation $g$, one has $g\circ f=\varnothing$.