Remark 57.5.3. Let $X$ be a scheme. Consider the natural functors $F_1 : \textit{FÉt}_ X \to \mathit{Sch}$ and $F_2 : \textit{FÉt}_ X \to \mathit{Sch}/X$. Then

1. The functors $F_1$ and $F_2$ commute with finite colimits.

2. The functor $F_2$ commutes with finite limits,

3. The functor $F_1$ commutes with connected finite limits, i.e., with equalizers and fibre products.

The results on limits are immediate from the discussion in the proof of Lemma 57.5.2 and Categories, Lemma 4.16.2. It is clear that $F_1$ and $F_2$ commute with finite coproducts. By the dual of Categories, Lemma 4.23.2 we need to show that $F_1$ and $F_2$ commute with coequalizers. In the proof of Lemma 57.5.2 we saw that coequalizers in $\textit{FÉt}_ X$ look étale locally like this

$\xymatrix{ \coprod _{j \in J} U \ar@<1ex>[r]^ a \ar@<-1ex>[r]_ b & \coprod _{i \in I} U \ar[r] & \coprod _{t \in \text{Coeq}(a, b)} U }$

which is certainly a coequalizer in the category of schemes. Hence the statement follows from the fact that being a coequalizer is fpqc local as formulated precisely in Descent, Lemma 35.10.8.

Comment #6533 by Tim Holzschuh on

Typo in the last sentence: "as formulate precisely" -- there's a 'd' missing in "formulated"

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