This result is a trivial consequence of Theorem \ref{theorem-cohomology-hypercoverings}. Namely, we can artificially replace $\mathcal{C}$ with a slightly bigger site $\mathcal{C}'$ such that (I) $\mathcal{C}'$ has a final object $X$ and (II) hypercoverings in $\mathcal{C}$ are more or less the same thing as hypercoverings of $X$ in $\mathcal{C}'$. But due to the nature of things, there is quite a bit of bookkeeping to do. \medskip\noindent Let us call a family of morphisms $\{U_i \to U\}$ in $\mathcal{C}$ with fixed target a {\it weak covering} if the sheafification of the map $\coprod_{i \in I} h_{U_i} \to h_U$ becomes surjective. We construct a new site $\mathcal{C}'$ as follows \begin{enumerate} \item as a category set $\Ob(\mathcal{C}') = \Ob(\mathcal{C}) \amalg \{X\}$ and add a unique morphism to $X$ from every object of $\mathcal{C}'$, \item $\mathcal{C}'$ has fibre products as fibre products and products of pairs exist in $\mathcal{C}$, \item coverings of $\mathcal{C}'$ are weak coverings of $\mathcal{C}$ together with those $\{U_i \to X\}_{i \in I}$ such that either $U_i = X$ for some $i$, or $U_i \not = X$ for all $i$ and the map $\coprod h_{U_i} \to *$ of presheaves on $\mathcal{C}$ becomes surjective after sheafification on $\mathcal{C}$, \item we apply Sets, Lemma \ref{sets-lemma-coverings-site} to restrict the coverings to obtain our site $\mathcal{C}'$. \end{enumerate} Then $\Sh(\mathcal{C}') = \Sh(\mathcal{C})$ because the inclusion functor $\mathcal{C} \to \mathcal{C}'$ is a special cocontinuous functor (see Sites, Definition \ref{sites-definition-special-cocontinuous-functor}). We omit the straightforward verifications. \medskip\noindent Choose a covering $\{U_i \to X\}$ of $\mathcal{C}'$ such that $U_i$ is an object of $\mathcal{C}$ for all $i$ (possible because $\mathcal{C} \to \mathcal{C}'$ is special cocontinuous). Then $K_0 = \{U_i \to X\}$ is a covering in the site $\mathcal{C}'$ constructed above. We view $K_0$ as an object of $\text{SR}(\mathcal{C}', X)$ and we set $K_{init} = \text{cosk}_0(K_0)$. Then $K_{init}$ is a hypercovering of $X$, see Example \ref{example-cech}. Note that every $K_{init, n}$ has the shape $\{W_j \to X\}$ with $W_j \in \Ob(\mathcal{C})$. \medskip\noindent Proof of (1). Choose $\xi \in H^i(\mathcal{F}) = H^i(X, \mathcal{F}')$ where $\mathcal{F}'$ is the abelian sheaf on $\mathcal{C}'$ corresponding to $\mathcal{F}$ on $\mathcal{C}$. By Theorem \ref{theorem-cohomology-hypercoverings} there exists a morphism of hypercoverings $K' \to K_{init}$ of $X$ in $\mathcal{C}'$ such that $\xi$ comes from an element of $\check{H}^i(K', \mathcal{F})$. Write $K'_n = \{U_{n, j} \to X\}$. Now since $K'_n$ maps to $K_{init, n}$ we see that $U_{n, j}$ is an object of $\mathcal{C}$. Hence we can define a simplicial object $K$ of $\text{SR}(\mathcal{C})$ by setting $K_n = \{U_{n, j}\}$. Since coverings in $\mathcal{C}'$ consisting of families of morphisms of $\mathcal{C}$ are weak coverings, we see that $K$ is a hypercovering in the sense of Definition \ref{definition-hypercovering-variant}. Finally, since $\mathcal{F}'$ is the unique sheaf on $\mathcal{C}'$ whose restriction to $\mathcal{C}$ is equal to $\mathcal{F}$ we see that the {\v C}ech complexes $s(\mathcal{F}(K))$ and $s(\mathcal{F}'(K'))$ are identical and (1) follows. (Compatibility with map into cohomology groups omitted.) \medskip\noindent Proof of (2). Let $K$ and $L$ be hypercoverings in $\mathcal{C}$. Let $K'$ and $L'$ be the simplicial objects of $\text{SR}(\mathcal{C}', X)$ gotten from $K$ and $L$ by the functor $\text{SR}(\mathcal{C}) \to \text{SR}(\mathcal{C}', X)$, $\{U_i\} \mapsto \{U_i \to X\}$. As before we have equality of {\v C}ech complexes and hence we obtain $\xi_{K'}$ and $\xi_{L'}$ mapping to the same cohomology class of $\mathcal{F}'$ over $\mathcal{C}'$. After possibly enlarging our choice of coverings in $\mathcal{C}'$ (due to a set theoretical issue) we may assume that $K'$ and $L'$ are hypercoverings of $X$ in $\mathcal{C}'$; this is true by our definition of hypercoverings in Definition \ref{definition-hypercovering-variant} and the fact that weak coverings in $\mathcal{C}$ give coverings in $\mathcal{C}'$. By Theorem \ref{theorem-cohomology-hypercoverings} there exists a hypercovering $M'$ of $X$ in $\mathcal{C}'$ and morphisms $M' \to K'$, $M' \to L'$, and $M' \to K_{init}$ such that $\xi_{K'}$ and $\xi_{L'}$ restrict to the same element of $\check{H}^i(M', \mathcal{F})$. Unwinding this statement as above we find that (2) is true.