Proposition 4.37.
A morphism of logoi \(\phi^*\colon T \to T'\) is étale if and only if:
The functor \(\phi^*\) admits a left adjoint \(\phi_{\sharp}\).
The functor \(\phi_{\sharp}\) is conservative.
For every morphism \(X\to Y\) in \(T\) and every morphism \(Z\to\phi^*Y\) in \(T'\), the canonical map
\[\phi_{\sharp}\bigl(\phi^*X \times_{\phi^*Y} Z\bigr) \longrightarrow X \times_Y \phi_{\sharp}(Z)\]is an isomorphism.
Proof
If \(\phi^*\) is étale, choose \(A\in T\) and an identification \(T'\simeq T_{/A}\) under which \(\phi^*\) is the functor \(U\mapsto U\times A\). Its left adjoint \(\phi_{\sharp}\) is the forgetful functor \(T_{/A}\to T\). It is conservative, and the projection formula follows by composing pullback squares in \(T\).Conversely, suppose that \(\phi^*\) admits a conservative left adjoint \(\phi_{\sharp}\) satisfying the projection formula, and set \(A:=\phi_{\sharp}(*)\). Consider the functor
\[F\colon T' \longrightarrow T_{/A}, \qquad W \longmapsto \bigl(\phi_{\sharp}(W)\to A\bigr).\]
It admits a right adjoint \[G(Y\to A):=\phi^*(Y)\times_{\phi^*A}*.\]
For \(Y\to A\), the projection formula identifies the underlying morphism of the counit \(FG(Y)\to Y\) with an isomorphism, so the counit is an isomorphism in \(T_{/A}\). Let \(W\to GFW\) be the unit. By the triangle identity, its image under \(F\) is an isomorphism because the counit is. In particular, \(\phi_{\sharp}\) sends the unit to an isomorphism. Since \(\phi_{\sharp}\) is conservative, the unit itself is an isomorphism. Thus \(F\) is an equivalence. Under this equivalence, \(\phi^*\) sends \(Y\) to the projection \(Y\times A\to A\), as required.