The Stacks project

Lemma 80.4.5. Let $S$ be a scheme. Let $F, G : (\mathit{Sch}/S)_{fppf}^{opp} \to \textit{Sets}$. Let $a : F \to G$ be a transformation of functors representable by algebraic spaces. Let $\mathcal{P}$, $\mathcal{P}'$ be properties as in Definition 80.4.1. Suppose that for any morphism $f : X \to Y$ of algebraic spaces over $S$ we have $\mathcal{P}(f) \Rightarrow \mathcal{P}'(f)$. If $a$ has property $\mathcal{P}$, then $a$ has property $\mathcal{P}'$.