I’ve added the following to my research book:
Definition
Galois surjection is the special case of Galois connection such that $latex f^{\ast} \circ f_{\ast} $ is identity.
Proposition
For Galois surjection $latex \mathfrak{A} \rightarrow \mathfrak{B}$ such that $latex \mathfrak{A}$ is a join-semilattice we have (for every $latex y \in \mathfrak{B}$)
$latex f_{\ast} y = \max \{ x \in \mathfrak{A} \mid f^{\ast} x = y \}.$
(Don’t confuse this my little theorem with the well-known theorem with similar formula formula $latex f_{\ast} y = \max \{ x \in \mathfrak{A} \mid f^{\ast} x \leq y \}$.)
This formula in particular applies to the Galois connection between funcoids and reloids (see my book).