Theorem. the classifying property of BG [fgap-000R]
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    Let \(G\) be a group and \(\mathcal {E}\) a topos over \(\mathsf {Set}\). There is a natural equivalence \[ \operatorname {Geom}_{/\mathsf {Set}}(\mathcal {E},\mathsf {B}G) \simeq \operatorname {Tor}(\mathcal {E},G) \] between geometric morphisms over \(\mathsf {Set}\) to \(\mathsf {B}G\) and \(G\)-torsors in \(\mathcal {E}\). Under this equivalence, a geometric morphism \[ f:\mathcal {E}\longrightarrow \mathsf {B}G \] classifies the torsor \(f^*U_G\).

    This is Theorem VIII.2.7 of [maclane1992sheaves]; its theorem and proof are on printed pp. 431--433.

    The two directions are visible in the diagram

    The geometric morphism points toward the classifying topos. Its inverse-image functor carries the universal torsor back to the topos in which the classified torsor lives.

    Proof. 

      Inverse image preserves finite limits and, as a left adjoint, colimits. Every epimorphism in a topos is regular, so inverse image carries the universal torsor equations, including the epimorphic map to \(1\), to torsor equations in \(\mathcal {E}\). Conversely, the cited theorem constructs a geometric morphism from a torsor and proves that the two constructions are naturally inverse. The complete construction belongs to the classifying theorem; the point used here is its variance and universal property.

      For a single finite action, freeness and transitivity can be checked without topoi. The added value here is one universal object classifying torsors in every topos over \(\mathsf {Set}\) and carrying them along inverse image.