A function on binary relations

A function on binary relations

Let $\rho$ is a function mapping every binary relation $f$ (on some set
$U$) into a function which maps binary relations into binary relations by
the formula
$$(\rho(f))(g) = f\circ g.$$
Is $\rho$:
upper adjoint in a Galois connection?
lower adjoint in a Galois connection?
meet-semilattice homomorphisms?
join-semilattice homomorphisms?
(The order assumed is the set-theoretic inclusion of binary relations.)
If it has adjoints, what these adjoints are?