nLab
internally projective object

An object E of a topos is called internally projective if the functor () E preserves epimorphisms. Any projective object is internally projective. The internal axiom of choice (that is, the axiom of choice interpreted in the internal logic of the topos) is equivalent to the statement that every object is internally projective. This is strictly weaker than the “external” axiom of choice that every epimorphism in the topos is split.