The Stacks project

Lemma 18.19.5. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $f : V \to U$ be a morphism of $\mathcal{C}$. Then there exists a commutative diagram

\[ \xymatrix{ (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}/V), \mathcal{O}_ V) \ar[rd]_{(j_ V, j_ V^\sharp )} \ar[rr]_{(j, j^\sharp )} & & (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}/U), \mathcal{O}_ U) \ar[ld]^{(j_ U, j_ U^\sharp )} \\ & (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O}) & } \]

of ringed topoi. Here $(j, j^\sharp )$ is the localization morphism associated to the object $V/U$ of the ringed site $(\mathcal{C}/V, \mathcal{O}_ V)$.

Proof. The only thing to check is that $j_ V^\sharp = j^\sharp \circ j^{-1}(j_ U^\sharp )$, since everything else follows directly from Sites, Lemma 7.25.8 and Sites, Equation (7.25.8.1). We omit the verification of the equality. $\square$

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