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There is on the nLab page "Grothendieck topos" a part about the theory of Grothendieck toposes in weak foundation. It claims that the equivalence for a category between the Giraud's axioms and being ...

There are several theorems in category-theoretic logic which say something like, "any proposition in X logic that is provable in topos logic assuming (the law of excluded middle and) the axiom of ...