Lemma 7.31.1. Let $f : \mathop{\mathit{Sh}}\nolimits (\mathcal{C}) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{D})$ be a morphism of topoi. Let $\mathcal{G}$ be a sheaf on $\mathcal{D}$. Set $\mathcal{F} = f^{-1}\mathcal{G}$. Then there exists a commutative diagram of topoi
\[ \xymatrix{ \mathop{\mathit{Sh}}\nolimits (\mathcal{C})/\mathcal{F} \ar[r]_{j_\mathcal {F}} \ar[d]_{f'} & \mathop{\mathit{Sh}}\nolimits (\mathcal{C}) \ar[d]^ f \\ \mathop{\mathit{Sh}}\nolimits (\mathcal{D})/\mathcal{G} \ar[r]^{j_\mathcal {G}} & \mathop{\mathit{Sh}}\nolimits (\mathcal{D}). } \]
The morphism $f'$ is characterized by the property that
\[ (f')^{-1}(\mathcal{H} \xrightarrow {\varphi } \mathcal{G}) = (f^{-1}\mathcal{H} \xrightarrow {f^{-1}\varphi } \mathcal{F}) \]
and we have $f'_*j_\mathcal {F}^{-1} = j_\mathcal {G}^{-1}f_*$.
Proof.
Since the statement is about topoi and does not refer to the underlying sites we may change sites at will. Hence by the discussion in Remark 7.29.7 we may assume that $f$ is given by a continuous functor $u : \mathcal{D} \to \mathcal{C}$ satisfying the assumptions of Proposition 7.14.7 between sites having all finite limits and subcanonical topologies, and such that $\mathcal{G} = h_ V$ for some object $V$ of $\mathcal{D}$. Then $\mathcal{F} = f^{-1}h_ V = h_{u(V)}$ by Lemma 7.13.5. By Lemma 7.28.1 we obtain a commutative diagram of morphisms of topoi
\[ \xymatrix{ \mathop{\mathit{Sh}}\nolimits (\mathcal{C}/U) \ar[r]_{j_ U} \ar[d]_{f'} & \mathop{\mathit{Sh}}\nolimits (\mathcal{C}) \ar[d]^ f \\ \mathop{\mathit{Sh}}\nolimits (\mathcal{D}/V) \ar[r]^{j_ V} & \mathop{\mathit{Sh}}\nolimits (\mathcal{D}), } \]
and we have $f'_*j_ U^{-1} = j_ V^{-1}f_*$. By Lemma 7.30.5 we may identify $j_\mathcal {F}$ and $j_ U$ and $j_\mathcal {G}$ and $j_ V$. The description of $(f')^{-1}$ is given in Lemma 7.28.1.
$\square$
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