The functor \(\Topos\catop \to \widehat{\Cat}, T \mapsto \Topos^{\et}_{/T}\) is equivalent to the composite \(\Topos\catop \simeq \Logos \hookrightarrow \Cat \hookrightarrow \widehat{\Cat}\).
and \(T\to\Topos^{\et}_{/T}\), \(X\mapsto T_{/X}\). The classification in Proposition 4.38 supplies inverse equivalences. These equivalences are natural in \(T\): for a pullback square we have a canonical isomorphism \(\psi^*\phi_{\sharp}(*) \simeq \phi'_{\sharp}(*)\). These base-change isomorphisms are compatible with pasting, since both sides classify the same iterated pullback of the étale morphism \(\phi\). Thus the objectwise equivalences assemble into the asserted equivalence of functors.
