Let $\StrTop{\ETop}$ and $\StrTop{\FTop}$ be two topoi ringed in local $k$-algebras. A morphism from $\StrTop{\ETop}$ to $\StrTop{\FTop}$ is a geometric morphism $\phi : \ETop\to \FTop$ together with a morphism of $k$-algebras $f : \InvImg{\phi}\Str{\FTop}\to \Str{\ETop}$. This morphism will be “local” if for each elementary étalé $U\to V$ the square

is cartesian. We thus recover the local morphisms (without quotation marks) for the two localization data already considered.