**Proof.**
Let $\varphi : \mathcal{F} \to \mathcal{G}$ be a map of adequate $\mathcal{O}$-modules. To prove (1) and (2) it suffices to show that $\mathcal{K} = \mathop{\mathrm{Ker}}(\varphi )$ and $\mathcal{Q} = \mathop{\mathrm{Coker}}(\varphi )$ computed in $\textit{Mod}((\mathit{Sch}/S)_\tau , \mathcal{O})$ are adequate. Let $U = \mathop{\mathrm{Spec}}(A)$ be an affine scheme over $S$. Let $F = F_{\mathcal{F}, A}$ and $G = F_{\mathcal{G}, A}$. By Lemmas 46.3.11 and 46.3.10 the kernel $K$ and cokernel $Q$ of the induced map $F \to G$ are adequate functors. Because the kernel is computed on the level of presheaves, we see that $K = F_{\mathcal{K}, A}$ and we conclude $\mathcal{K}$ is adequate. To prove the result for the cokernel, denote $\mathcal{Q}'$ the presheaf cokernel of $\varphi $. Then $Q = F_{\mathcal{Q}', A}$ and $\mathcal{Q} = (\mathcal{Q}')^\# $. Hence $\mathcal{Q}$ is adequate by Lemma 46.5.10.

Let $0 \to \mathcal{F}_1 \to \mathcal{F}_2 \to \mathcal{F}_3 \to 0$ is a short exact sequence of $\mathcal{O}$-modules and $\mathcal{F}_1$ and $\mathcal{F}_3$ are adequate. Let $U = \mathop{\mathrm{Spec}}(A)$ be an affine scheme over $S$. Let $F_ i = F_{\mathcal{F}_ i, A}$. The sequence of functors

\[ 0 \to F_1 \to F_2 \to F_3 \to 0 \]

is exact, because for $V = \mathop{\mathrm{Spec}}(B)$ affine over $U$ we have $H^1(V, \mathcal{F}_1) = 0$ by Lemma 46.5.8. Since $F_1$ and $F_3$ are adequate functors by Lemma 46.5.2 we see that $F_2$ is adequate by Lemma 46.3.16. Thus $\mathcal{F}_2$ is adequate.

Let $\mathcal{I} \to \textit{Adeq}(\mathcal{O})$, $i \mapsto \mathcal{F}_ i$ be a diagram. Denote $\mathcal{F} = \mathop{\mathrm{colim}}\nolimits _ i \mathcal{F}_ i$ the colimit computed in $\textit{Mod}((\mathit{Sch}/S)_\tau , \mathcal{O})$. To prove (4) it suffices to show that $\mathcal{F}$ is adequate. Let $\mathcal{F}' = \mathop{\mathrm{colim}}\nolimits _ i \mathcal{F}_ i$ be the colimit computed in presheaves of $\mathcal{O}$-modules. Then $\mathcal{F} = (\mathcal{F}')^\# $. Let $U = \mathop{\mathrm{Spec}}(A)$ be an affine scheme over $S$. Let $F_ i = F_{\mathcal{F}_ i, A}$. By Lemma 46.3.12 the functor $\mathop{\mathrm{colim}}\nolimits _ i F_ i = F_{\mathcal{F}', A}$ is adequate. Lemma 46.5.10 shows that $\mathcal{F}$ is adequate.
$\square$

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