Definition 6.77. (Essential morphism)
A morphism of topoi \(\phi_* \colon T \to S\) is called essential if the inverse image functor \(\phi^*\colon S \to T\) admits a further left adjoint \(\phi_\sharp \colon T \to S\). We thus have an adjoint triple
\[\phi_\sharp \quad\dashv\quad \phi^* \quad\dashv\quad \phi_*.\]
We call a topos \(T\) essential if the terminal morphism \(\Gamma_*\colon T \to \An\) is essential.