Necessary and sufficient conditions are given for the Eilenberg-Moore comparison functor Φ \Phi arising from a functor U (having a left adjoint) to be a Galois connection in the sense of J. R. Isbell, in which case the functor U is said to be of subdescent type. These conditions, when applied to a contravariant hom-functor U = C ( − , B ) : C op → S e t U = {\mathbf {C}}( - ,B):{{\mathbf {C}}^{{\text {op}}}} \to {\mathbf {Set}} , read like a kind of functional completeness axiom for the object B. In order to appreciate this result, it is useful to consider the full subcategory d o m B ⊂ C {\mathbf {dom}_B} \subset {\mathbf {C}} of so-called B-dominions, consisting of certain canonically arising regular subobjects of powers of the object B. The functor U = C ( − , B ) U = {\mathbf {C}}( - ,B) is of subdescent type if and only if the object B is a regular cogenerator for the category d o m B {\mathbf {dom}_B} , in which case d o m B {\mathbf {dom}_B} is the reflective hull of B in C and, moreover, the category d o m B {\mathbf {dom}_B} admits a Stone-like representation as (being contravariantly equivalent, via the comparison functor Φ \Phi , to) a full, reflective subcategory of the category of algebras for the triple in Set induced by the functor U.