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.
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