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
\[ \xymatrix{ f^{-1}(\mathcal{O}^*_\mathcal {D}) \ar[r]_-{f^\sharp } \ar[d] & \mathcal{O}^*_\mathcal {C} \ar[d] \\ f^{-1}(\mathcal{O}_\mathcal {D}) \ar[r]^-{f^\sharp } & \mathcal{O}_\mathcal {C} } \]
(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.
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