The Stacks project

Proof. Recall that $\mathcal{F}^\# $ is equal to $(\mathcal{F}^+)^+$, see Sites, Theorem 7.10.10, where $\mathcal{F}^+$ is the presheaf which sends an object $U$ of $X_{pro\text{-}\acute{e}tale}$ to $\mathop{\mathrm{colim}}\nolimits H^0(\mathcal{U}, \mathcal{F})$ where the colimit is over all pro-étale coverings $\mathcal{U}$ of $U$. Thus it suffices to prove that (a) $\mathcal{F}^+$ sends finite disjoint unions to products and (b) sends $W$ to $\mathcal{F}(W)$. If $U = U_1 \amalg U_2$, then given a pro-étale covering $\mathcal{U} = \{ f_ j : V_ j \to U\} $ of $U$ we obtain pro-étale coverings $\mathcal{U}_ i = \{ f_ j^{-1}(U_ i) \to U_ i\} $ and we clearly have

because $\mathcal{F}$ sends finite disjoint unions to products (this includes the condition that $\mathcal{F}$ sends the empty scheme to the singleton). This proves (a). Finally, any pro-étale covering of $W$ can be refined by a finite disjoint union decomposition $W = W_1 \amalg \ldots W_ n$ by Lemma 61.13.2. Hence $\mathcal{F}^+(W) = \mathcal{F}(W)$ exactly because the value of $\mathcal{F}$ on $W$ is the product of the values of $\mathcal{F}$ on the $W_ j$. This proves (b). $\square$