Lemma 18.20.1. 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 $V$ be an object of $\mathcal{D}$ and set $U = u(V)$. Then there is a canonical map of sheaves of rings $(f')^\sharp $ such that the diagram of Sites, Lemma 7.28.1 is turned into a commutative diagram of ringed topoi

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

Moreover, in this situation we have $f'_*j_ U^{-1} = j_ V^{-1}f_*$ and $f'_*j_ U^* = j_ V^*f_*$.

**Proof.**
Just take $(f')^\sharp $ to be

\[ (f')^{-1}\mathcal{O}'_ V = (f')^{-1}j_ V^{-1}\mathcal{O}' = j_ U^{-1}f^{-1}\mathcal{O}' \xrightarrow {j_ U^{-1}f^\sharp } j_ U^{-1}\mathcal{O} = \mathcal{O}_ U \]

and everything else follows from Sites, Lemma 7.28.1. (Note that $j^{-1} = j^*$ on sheaves of modules if $j$ is a localization morphism, hence the first equality of functors implies the second.)
$\square$

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