The Stacks project

Lemma 7.31.4. 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 $\mathcal{D}$. Let $c : U \to u(V)$ be a morphism. Set $\mathcal{G} = h_ V^\# $ and $\mathcal{F} = h_ U^\# = f^{-1}h_ V^\# $. Let $s : \mathcal{F} \to f^{-1}\mathcal{G}$ be the map induced by $c$. Then the diagram of morphisms of topoi of Lemma 7.28.3 agrees with the diagram of morphisms of topoi of Lemma 7.31.3 via the identifications $j_\mathcal {F} = j_ U$ and $j_\mathcal {G} = j_ V$ of Lemma 7.30.5.

Proof. This follows on combining Lemmas 7.30.7 and 7.31.2. $\square$

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