 as a way of getting rid of the complex substitution rule of Church (1956). It is clear that, in case the only requirement you impose to the universes of the second-order structure is to be subsets of standard universes, there is no guarantee that they are models of Comprehension. That is why we need the general structures to obey extra rules, universes have to be closed under definability. With general structures the set of validities coincides with the set of sentences derivable in the second-order calculus. So, we go back to the situation encountered in first-order logic as you recover: Compactness, Löwenheim-Skolem and Completeness.