It is well-known the the category $\mathcal Set$ is a topos. In this topos, how would one translate the Axiom schema of Separation into purely category-theoretic terms? I ask this question because of the following theorem of Radu Diaconescu (found in his paper "Axiom of Choice and Complementation", Proceedings of the American Mathematical Society, Volume 51, Number 1, August 1975):
$AC$ implies that every subobject has a complement
Though this may show my ignorance regarding both the Axiom schema of Separation and Category theory, a cursory reading of Separation seems to suggest that, at least for subsets of some set $\mathcal S$ defined by some formula $\varphi$ in the language of set theory, these subsets have relative complements defined by the formula $\lnot$$\varphi$.
Also, in his paper, Diaconescu has the following theorem (from which the theorem stated above follows as a corollary. In stating this theorem, I will replace his chosen symbol $\mathscr E$ for an arbitrary topos with the topos of my interest $\mathcal Set$:
Any coequalizer of two nonintersecting monomorphisms has a section iff in $\mathcal Set$ subobjects have complements. (Perhaps this theorem might give readers a clue as to how to translate the axiom schema of Separation into purely category-theory terms.)
I am hoping by this to discover relations betwen the axiom schema of Separation and the Axiom of Choice (or, in the alternative, that no relation exists).
Thanks in advance for any help given.
(Addendum) Since Stefan, in his comment to me, raised the issue of the distinctness between the set-theoretic definition of complement and the category-theoretic definition of complement, I now ask the following question(s):
In the category (Topos) $\mathcal Set$: in what ways does the category-theoretic definition of complement coincide with the set-theoretic definition of complement (if they coincide at all)? In what ways do they differ (if, in fact, they differ at all)?