The Stacks project

Lemma 46.5.4. Let $f : T \to S$ be a morphism of schemes. The pullback $f^*\mathcal{F}$ of an adequate $\mathcal{O}$-module $\mathcal{F}$ on $(\mathit{Sch}/S)_\tau $ is an adequate $\mathcal{O}$-module on $(\mathit{Sch}/T)_\tau $.

Proof. The pullback map $f^* : \textit{Mod}((\mathit{Sch}/S)_\tau , \mathcal{O}) \to \textit{Mod}((\mathit{Sch}/T)_\tau , \mathcal{O})$ is given by restriction, i.e., $f^*\mathcal{F}(V) = \mathcal{F}(V)$ for any scheme $V$ over $T$. Hence this lemma follows immediately from Lemma 46.5.2 and the definition. $\square$

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