Situation 99.3.1. Let $S$ be a scheme. Let $f : X \to B$ be a morphism of algebraic spaces over $S$. Let $\mathcal{F}$, $\mathcal{G}$ be quasi-coherent $\mathcal{O}_ X$-modules. For any scheme $T$ over $B$ we will denote $\mathcal{F}_ T$ and $\mathcal{G}_ T$ the base changes of $\mathcal{F}$ and $\mathcal{G}$ to $T$, in other words, the pullbacks via the projection morphism $X_ T = X \times _ B T \to X$. We consider the functor

99.3.1.1

\begin{equation} \label{quot-equation-hom} \mathit{Hom}(\mathcal{F}, \mathcal{G}) : (\mathit{Sch}/B)^{opp} \longrightarrow \textit{Sets},\quad T \longrightarrow \mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}_{X_ T}}(\mathcal{F}_ T, \mathcal{G}_ T) \end{equation}

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