Proof of the results on $\text{SR}(\mathcal{C})$. Proof of (1). The coproduct of $\{U_i\}_{i \in I}$ and $\{V_j\}_{j \in J}$ is $\{U_i\}_{i \in I} \amalg \{V_j\}_{j \in J}$, in other words, the family of objects whose index set is $I \amalg J$ and for an element $k \in I \amalg J$ gives $U_i$ if $k = i \in I$ and gives $V_j$ if $k = j \in J$. Similarly for coproducts of families of objects. It is clear that $F$ commutes with these. \medskip\noindent Proof of (2). For $U$ in $\Ob(\mathcal{C})$ consider the object $\{U\}$ of $\text{SR}(\mathcal{C})$. It is clear that $\Mor_{\text{SR}(\mathcal{C})}(\{U\}, K)) = F(K)(U)$ for $K \in \Ob(\text{SR}(\mathcal{C}))$. Since limits of presheaves are computed at the level of sections (Sites, Section \ref{sites-section-limits-colimits-PSh}) we conclude that $F$ commutes with limits. \medskip\noindent Proof of (3). Suppose given a morphism $(\alpha, f_i) : \{U_i\}_{i \in I} \to \{V_j\}_{j \in J}$ and a morphism $(\beta, g_k) : \{W_k\}_{k \in K} \to \{V_j\}_{j \in J}$. The fibred product of these morphisms is given by $$ \{ U_i \times_{f_i, V_j, g_k} W_k\}_{(i, j, k) \in I \times J \times K \text{ such that } j = \alpha(i) = \beta(k)} $$ The fibre products exist if $\mathcal{C}$ has fibre products. \medskip\noindent Proof of (4). The product of $\{U_i\}_{i \in I}$ and $\{V_j\}_{j \in J}$ is $\{U_i \times V_j\}_{i \in I, j \in J}$. The products exist if $\mathcal{C}$ has products. \medskip\noindent Proof of (5). The equalizer of two maps $(\alpha, f_i), (\alpha', f'_i) : \{U_i\}_{i \in I} \to \{V_j\}_{j \in J}$ is $$ \{ \text{Eq}(f_i, f'_i : U_i \to V_{\alpha(i)}) \}_{i \in I,\ \alpha(i) = \alpha'(i)} $$ The equalizers exist if $\mathcal{C}$ has equalizers. \medskip\noindent Proof of (6). If $X$ is a final object of $\mathcal{C}$, then $\{X\}$ is a final object of $\text{SR}(\mathcal{C})$. \medskip\noindent Proof of the statements about $\text{SR}(\mathcal{C}, X)$. These follow from the results above applied to the category $\mathcal{C}/X$ using that $\text{SR}(\mathcal{C}/X) = \text{SR}(\mathcal{C}, X)$ and that $\textit{PSh}(\mathcal{C}/X) = \textit{PSh}(\mathcal{C})/h_X$ (Sites, Lemma \ref{sites-lemma-essential-image-j-shriek} applied to $\mathcal{C}$ endowed with the chaotic topology). However we also argue directly as follows. It is clear that the coproduct of $\{U_i \to X\}_{i \in I}$ and $\{V_j \to X\}_{j \in J}$ is $\{U_i \to X\}_{i \in I} \amalg \{V_j \to X\}_{j \in J}$ and similarly for coproducts of families of families of morphisms with target $X$. The object $\{X \to X\}$ is a final object of $\text{SR}(\mathcal{C}, X)$. Suppose given a morphism $(\alpha, f_i) : \{U_i \to X\}_{i \in I} \to \{V_j \to X\}_{j \in J}$ and a morphism $(\beta, g_k) : \{W_k \to X\}_{k \in K} \to \{V_j \to X\}_{j \in J}$. The fibred product of these morphisms is given by $$ \{ U_i \times_{f_i, V_j, g_k} W_k \to X \}_{(i, j, k) \in I \times J \times K \text{ such that } j = \alpha(i) = \beta(k)} $$ The fibre products exist by the assumption that $\mathcal{C}$ has fibre products. Thus $\text{SR}(\mathcal{C}, X)$ has finite limits, see Categories, Lemma \ref{categories-lemma-finite-limits-exist}. We omit verifying the statements on the functor $F$ in this case.