Situation 77.7.1. Let $S$ be a scheme. Let $f : X \to B$ be a morphism of algebraic spaces over $S$. Let $u : \mathcal{F} \to \mathcal{G}$ be a homomorphism of quasi-coherent $\mathcal{O}_ X$-modules. For any scheme $T$ over $B$ we will denote $u_ T : \mathcal{F}_ T \to \mathcal{G}_ T$ the base change of $u$ to $T$, in other words, $u_ T$ is the pullback of $u$ via the projection morphism $X_ T = X \times _ B T \to X$. In this situation we can consider the functor
77.7.1.1
\begin{equation} \label{spaces-flat-equation-iso} F_{iso} : (\mathit{Sch}/B)^{opp} \longrightarrow \textit{Sets}, \quad T \longrightarrow \left\{ \begin{matrix} \{ *\}
& \text{if }u_ T\text{ is an isomorphism},
\\ \emptyset
& \text{else.}
\end{matrix} \right. \end{equation}
There are variants $F_{inj}$, $F_{surj}$, $F_{zero}$ where we ask that $u_ T$ is injective, surjective, or zero.
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