Lemma 7.30.6. Let $\mathcal{C}$ be a site. If $s : \mathcal{G} \to \mathcal{F}$ is a morphism of sheaves on $\mathcal{C}$ then there exists a natural commutative diagram of morphisms of topoi

\[ \xymatrix{ \mathop{\mathit{Sh}}\nolimits (\mathcal{C})/\mathcal{G} \ar[rd]_{j_\mathcal {G}} \ar[rr]_ j & & \mathop{\mathit{Sh}}\nolimits (\mathcal{C})/\mathcal{F} \ar[ld]^{j_\mathcal {F}} \\ & \mathop{\mathit{Sh}}\nolimits (\mathcal{C}) & } \]

where $j = j_{\mathcal{G}/\mathcal{F}}$ is the localization of the topos $\mathop{\mathit{Sh}}\nolimits (\mathcal{C})/\mathcal{F}$ at the object $\mathcal{G}/\mathcal{F}$. In particular we have

\[ j^{-1}(\mathcal{H} \to \mathcal{F}) = (\mathcal{H} \times _\mathcal {F} \mathcal{G} \to \mathcal{G}) \]

and

\[ j_!(\mathcal{E} \xrightarrow {e} \mathcal{F}) = (\mathcal{E} \xrightarrow {s \circ e} \mathcal{G}). \]

**Proof.**
The description of $j^{-1}$ and $j_!$ comes from the description of those functors in Lemma 7.30.1. The equality of functors $j_{\mathcal{G}!} = j_{\mathcal{F}!} \circ j_!$ is clear from the description of these functors (as forgetful functors). By adjointness we also obtain the equalities $j_\mathcal {G}^{-1} = j^{-1} \circ j_\mathcal {F}^{-1}$, and $j_{\mathcal{G}, *} = j_{\mathcal{F}, *} \circ j_*$.
$\square$

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