The Stacks project

Proof. Assume $F$ is limit preserving. It suffices to show that $F^+$ is limit preserving, since $F^\# = (F^+)^+$, see Sites, Theorem 7.10.10. Let $T$ be an affine scheme over $S$, and let $T = \mathop{\mathrm{lim}}\nolimits T_ i$ be written as the directed limit of an inverse system of affine $S$ schemes. Recall that $F^+(T)$ is the colimit of $\check H^0(\mathcal{V}, F)$ where the limit is over all coverings of $T$ in $(\mathit{Sch}/S)_{fppf}$. Any fppf covering of an affine scheme can be refined by a standard fppf covering, see Topologies, Lemma 34.7.4. Hence we can write

Any $\mathcal{V} = \{ T_ k \to T\} _{k = 1, \ldots , n}$ in the colimit may be written as $V_ i \times _{T_ i} T$ for some $i$ and some standard fppf covering $\mathcal{V}_ i = \{ T_{i, k} \to T_ i\} _{k = 1, \ldots , n}$ of $T_ i$. Denote $\mathcal{V}_{i'} = \{ T_{i', k} \to T_{i'}\} _{k = 1, \ldots , n}$ the base change for $i' \geq i$. Then we see that

Here the second equality holds because filtered colimits are exact. The third equality holds because $F$ is limit preserving and because $\mathop{\mathrm{lim}}\nolimits _{i' \geq i} T_{i', k} = T_ k$ and $\mathop{\mathrm{lim}}\nolimits _{i' \geq i} T_{i', k} \times _{T_{i'}} T_{i', l} = T_ k \times _ T T_ l$ by Limits, Lemma 32.2.3. If we use this for all coverings at the same time we obtain

The switch of the order of the colimits is allowed by Categories, Lemma 4.14.10. $\square$