Let $\phi$ be a function from a poset $B$ to a poset $A$.
$f \mapsto \min \{ g\in B \mid \phi(g) \geq f \}$ is called the lower adjoint of $\phi$ and $\phi$ is called an upper adjoint. These two functions together are called a Galois connection.
Is there any term for $f \mapsto \inf \{ g\in B \mid \phi(g) \geq f \}$ (specifically in the case if the lower adjoint does not exist and in the case if $B$ is a complete lattice)?
I've asked only about terminology, not about properties. However may properties be also interesting.