Lemma 27.4.1. In Situation 27.3.1. Let F be the functor associated to (S, \mathcal{A}) above. Let g : S' \to S be a morphism of schemes. Set \mathcal{A}' = g^*\mathcal{A}. Let F' be the functor associated to (S', \mathcal{A}') above. Then there is a canonical isomorphism
F' \cong h_{S'} \times _{h_ S} F
of functors.
Proof.
A pair (f' : T \to S', \varphi ' : (f')^*\mathcal{A}' \to \mathcal{O}_ T) is the same as a pair (f, \varphi : f^*\mathcal{A} \to \mathcal{O}_ T) together with a factorization of f as f = g \circ f'. Namely with this notation we have (f')^* \mathcal{A}' = (f')^*g^*\mathcal{A} = f^*\mathcal{A}. Hence the lemma.
\square
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