Lemma 18.20.2. Let $(f, f^\sharp ) : (\mathcal{C}, \mathcal{O}) \longrightarrow (\mathcal{D}, \mathcal{O}')$ be a morphism of ringed sites where $f$ is given by the continuous functor $u : \mathcal{D} \to \mathcal{C}$. Let $V \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{D})$, $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ and $c : U \to u(V)$ a morphism of $\mathcal{C}$. There exists a commutative diagram of ringed topoi

$\xymatrix{ (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}/U), \mathcal{O}_ U) \ar[rr]_{(j_ U, j_ U^\sharp )} \ar[d]_{(f_ c, f_ c^\sharp )} & & (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O}) \ar[d]^{(f, f^\sharp )} \\ (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}/V), \mathcal{O}'_ V) \ar[rr]^{(j_ V, j_ V^\sharp )} & & (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}), \mathcal{O}'). }$

The morphism $(f_ c, f_ c^\sharp )$ is equal to the composition of the morphism

$(f', (f')^\sharp ) : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}/u(V)), \mathcal{O}_{u(V)}) \longrightarrow (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}/V), \mathcal{O}'_ V)$

of Lemma 18.20.1 and the morphism

$(j, j^\sharp ) : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}/U), \mathcal{O}_ U) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}/u(V)), \mathcal{O}_{u(V)})$

of Lemma 18.19.5. Given any morphisms $b : V' \to V$, $a : U' \to U$ and $c' : U' \to u(V')$ such that

$\xymatrix{ U' \ar[r]_-{c'} \ar[d]_ a & u(V') \ar[d]^{u(b)} \\ U \ar[r]^-c & u(V) }$

commutes, then the following diagram of ringed topoi

$\xymatrix{ (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}/U'), \mathcal{O}_{U'}) \ar[rr]_{(j_{U'/U}, j_{U'/U}^\sharp )} \ar[d]_{(f_{c'}, f_{c'}^\sharp )} & & (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}/U), \mathcal{O}_ U) \ar[d]^{(f_ c, f_ c^\sharp )} \\ (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}/V'), \mathcal{O}'_{V'}) \ar[rr]^{(j_{V'/V}, j_{V'/V}^\sharp )} & & (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}/V), \mathcal{O}'_{V'}) }$

commutes.

Proof. On the level of morphisms of topoi this is Sites, Lemma 7.28.3. To check that the diagrams commute as morphisms of ringed topoi use Lemmas 18.19.5 and 18.20.1 exactly as in the proof of Sites, Lemma 7.28.3. $\square$

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