[1] Adjunction refers to an elementary operation on two sets, and has no bearing on the use of that term elsewhere in mathematics, including in category theory.
Montague (1961) showed that ZFC is not finitely axiomatizable, and his argument carries over to GST.
With its simple axioms, GST is also immune to the three great antinomies of naïve set theory: Russell's, Burali-Forti's, and Cantor's.
Given Adjunction, the usual construction of the successor ordinals from the empty set can proceed, one in which the natural numbers are defined as
GST is mutually interpretable with Peano arithmetic (thus it has the same proof-theoretic strength as PA).
The most remarkable fact about ST (and hence GST), is that these tiny fragments of set theory give rise to such rich metamathematics.
[2][3] This includes GST and every axiomatic set theory worth thinking about, assuming these are consistent.
Any axiomatizable theory, such as ST and GST, whose theorems include the Q axioms is likewise incomplete.
Given any model M of ZFC, the collection of hereditarily finite sets in M will satisfy the GST axioms.
Hence GST cannot ground analysis and geometry, and is too weak to serve as a foundation for mathematics.
Boolos was interested in GST only as a fragment of Z that is just powerful enough to interpret Peano arithmetic.
He never lingered over GST, only mentioning it briefly in several papers discussing the systems of Frege's Grundlagen and Grundgesetze, and how they could be modified to eliminate Russell's paradox.
The system Aξ'[δ0] in Tarski and Givant (1987: 223) is essentially GST with an axiom schema of induction replacing Specification, and with the existence of an empty set explicitly assumed.
[5] Burgess's theory ST[6] is GST with Empty Set replacing the axiom schema of specification.