The Stacks project

Lemma 96.17.1. Let $f : \mathcal{X} \to \mathcal{Y}$ be a $1$-morphism of categories fibred in groupoids over $(\mathit{Sch}/S)_{fppf}$. Let $\tau \in \{ Zar, {\acute{e}tale}, smooth, syntomic, fppf\} $.

1. $f_*\mathcal{I}$ is injective in $\textit{Ab}(\mathcal{Y}_\tau )$ for $\mathcal{I}$ injective in $\textit{Ab}(\mathcal{X}_\tau )$, and

2. $f_*\mathcal{I}$ is injective in $\textit{Mod}(\mathcal{Y}_\tau , \mathcal{O}_\mathcal {Y})$ for $\mathcal{I}$ injective in $\textit{Mod}(\mathcal{X}_\tau , \mathcal{O}_\mathcal {X})$.