Lemma 4.55.
Let \(\phi^*\colon T \to S\) be a morphism of logoi with a \(T\)-indexed left adjoint \(\phi_{\sharp}\). Then \(\phi^*\) factors as
\[T \xrightarrow{(-)\times \phi_{\sharp}(*)} T_{/\phi_{\sharp}(*)} \xrightarrow{\psi^*} S,\]
where \(\psi^*\) is fully faithful.
Proof
Let \(\eta\colon \id_S \to \phi^* \phi_{\sharp}\) be the unit of the adjunction \(\phi_{\sharp} \dashv \phi^*\). Define
\[\psi^*\colon T_{/\phi_{\sharp}(*)} \longrightarrow S\]
as the composite \[T_{/\phi_{\sharp}(*)} \xrightarrow{\phi^*} S_{/\phi^* \phi_{\sharp}(*)} \xrightarrow{\eta_*^*} S,\]
where \(\eta_*\colon * \to \phi^* \phi_{\sharp}(*)\) is the unit at the terminal object of \(S\) and the second functor is pullback along \(\eta_*\). For \(X\in T\), the object \(\psi^*(X\times \phi_{\sharp}(*))\) is the pullback of \(\phi^*(X)\times \phi^* \phi_{\sharp}(*) \to \phi^* \phi_{\sharp}(*)\) along \(\eta_*\), hence is equivalent to \(\phi^*(X)\).The left adjoint of \(\psi^*\) is \(\phi_{\sharp}\) followed by the evident map to \(\phi_{\sharp}(*)\). The counit is identified, by the \(T\)-indexed condition, with the transpose of a pullback square, and is therefore an isomorphism. Hence \(\psi^*\) is fully faithful.