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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