The Stacks project

Definition 18.40.9. Let $(f, f^\sharp ) : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O}_\mathcal {C}) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}), \mathcal{O}_\mathcal {D})$ be a morphism of ringed topoi. Assume $(\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O}_\mathcal {C})$ and $(\mathop{\mathit{Sh}}\nolimits (\mathcal{D}), \mathcal{O}_\mathcal {D})$ are locally ringed topoi. We say that $(f, f^\sharp )$ is a morphism of locally ringed topoi if and only if the diagram of sheaves

(see Lemma 18.40.8) is cartesian. If $(f, f^\sharp )$ is a morphism of ringed sites, then we say that it is a morphism of locally ringed sites if the associated morphism of ringed topoi is a morphism of locally ringed topoi.