Lemma 69.3.9. Let $S$ be a scheme contained in $\mathit{Sch}_{fppf}$. Let $F, G : (\mathit{Sch}/S)_{fppf}^{opp} \to \textit{Sets}$ be functors. If $a : F \to G$ is a transformation which is limit preserving, then the induced transformation of sheaves $F^\# \to G^\#$ is limit preserving.

Proof. Suppose that $T$ is a scheme and $y \in G^\# (T)$. We have to show the functor $F^\# _ y : (\mathit{Sch}/T)_{fppf}^{opp} \to \textit{Sets}$ constructed from $F^\# \to G^\#$ and $y$ as in Definition 69.3.1 is limit preserving. By Equation (69.3.1.1) we see that $F^\# _ y$ is a sheaf. Choose an fppf covering $\{ V_ j \to T\} _{j \in J}$ such that $y|_{V_ j}$ comes from an element $y_ j \in F(V_ j)$. Note that the restriction of $F^\#$ to $(\mathit{Sch}/V_ j)_{fppf}$ is just $F^\# _{y_ j}$. If we can show that $F^\# _{y_ j}$ is limit preserving then Lemma 69.3.8 guarantees that $F^\# _ y$ is limit preserving and we win. This reduces us to the case $y \in G(T)$.

Let $y \in G(T)$. In this case we claim that $F^\# _ y = (F_ y)^\#$. This follows from Equation (69.3.1.1). Thus this case follows from Lemma 69.3.7. $\square$

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