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).