Definition 4.1.
Given two topoi \(S\) and \(T\), a morphism of topoi (or geometric morphism) from \(S\) to \(T\) is a functor \(\phi_*\colon S \to T\) which admits a left exact colimit-preserving left adjoint \(\phi^*\colon T \to S\). We denote by
\[\Topos \quad \subseteq \quad \Cat\]
the subcategory spanned by the topoi and the morphisms of topoi.