Lemma 7.31.2. Let $f : \mathcal{C} \to \mathcal{D}$ be a morphism of sites given by the continuous functor $u : \mathcal{D} \to \mathcal{C}$. Let $V$ be an object of $\mathcal{D}$. Set $U = u(V)$. Set $\mathcal{G} = h_ V^\#$, and $\mathcal{F} = h_ U^\# = f^{-1}h_ V^\#$ (see Lemma 7.13.5). Then the diagram of morphisms of topoi of Lemma 7.31.1 agrees with the diagram of morphisms of topoi of Lemma 7.28.1 via the identifications $j_\mathcal {F}= j_ U$ and $j_\mathcal {G} = j_ V$ of Lemma 7.30.5.

Proof. This is not a complete triviality as the choice of morphism of sites giving rise to $f$ made in the proof of Lemma 7.31.1 may be different from the morphisms of sites given to us in the lemma. But in both cases the functor $(f')^{-1}$ is described by the same rule. Hence they agree and the associated morphism of topoi is the same. Some details omitted. $\square$

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