Lemma 35.10.6. Let S be a scheme.
The category \mathit{QCoh}((\mathit{Sch}/S)_{fppf}, \mathcal{O}) has colimits and they agree with colimits in the categories \textit{Mod}((\mathit{Sch}/S)_{Zar}, \mathcal{O}), \textit{Mod}((\mathit{Sch}/S)_{\acute{e}tale}, \mathcal{O}), and \textit{Mod}((\mathit{Sch}/S)_{fppf}, \mathcal{O}).
Given \mathcal{F}, \mathcal{G} in \mathit{QCoh}((\mathit{Sch}/S)_{fppf}, \mathcal{O}) the tensor products \mathcal{F} \otimes _\mathcal {O} \mathcal{G} computed in \textit{Mod}((\mathit{Sch}/S)_{Zar}, \mathcal{O}), \textit{Mod}((\mathit{Sch}/S)_{\acute{e}tale}, \mathcal{O}), or \textit{Mod}((\mathit{Sch}/S)_{fppf}, \mathcal{O}) agree and the common value is an object of \mathit{QCoh}((\mathit{Sch}/S)_{fppf}, \mathcal{O}).
Given \mathcal{F}, \mathcal{G} in \mathit{QCoh}((\mathit{Sch}/S)_{fppf}, \mathcal{O}) with \mathcal{F} finite locally free (in fppf, or equivalently étale, or equivalently Zariski topology) the internal homs \mathop{\mathcal{H}\! \mathit{om}}\nolimits _\mathcal {O}(\mathcal{F}, \mathcal{G}) computed in \textit{Mod}((\mathit{Sch}/S)_{Zar}, \mathcal{O}), \textit{Mod}((\mathit{Sch}/S)_{\acute{e}tale}, \mathcal{O}), or \textit{Mod}((\mathit{Sch}/S)_{fppf}, \mathcal{O}) agree and the common value is an object of \mathit{QCoh}((\mathit{Sch}/S)_{fppf}, \mathcal{O}).
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