I’ve added the following to my research book:
Galois surjection is the special case of Galois connection such that is identity.
For Galois surjection such that is a join-semilattice we have (for every )
(Don’t confuse this my little theorem with the well-known theorem with similar formula formula .)
This formula in particular applies to the Galois connection between funcoids and reloids (see my book).