Definition 70.3.1. Let $S$ be a scheme.
A functor $F : (\mathit{Sch}/S)_{fppf}^{opp} \to \textit{Sets}$ is said to be limit preserving or locally of finite presentation if for every affine scheme $T$ over $S$ which is a limit $T = \mathop{\mathrm{lim}}\nolimits T_ i$ of a directed inverse system of affine schemes $T_ i$ over $S$, we have
\[ F(T) = \mathop{\mathrm{colim}}\nolimits F(T_ i). \]We sometimes say that $F$ is locally of finite presentation over $S$.
Let $F, G : (\mathit{Sch}/S)_{fppf}^{opp} \to \textit{Sets}$. A transformation of functors $a : F \to G$ is limit preserving or locally of finite presentation if for every scheme $T$ over $S$ and every $y \in G(T)$ the functor
\[ F_ y : (\mathit{Sch}/T)_{fppf}^{opp} \longrightarrow \textit{Sets}, \quad T'/T \longmapsto \{ x \in F(T') \mid a(x) = y|_{T'}\} \]is locally of finite presentation over $T$1. We sometimes say that $F$ is relatively limit preserving over $G$.
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