We will prove that $c_{f_1, \ldots, f_c}(s) = \det(a_{ij}) c_{g_1, \ldots, g_c}(s)$ as global sections of $\mathcal{H}_Z(\mathcal{F})$ for any $s \in \mathcal{F}(X)$. This is sufficient since we then obtain the same result for section over any open subscheme of $X$. To do this, for $1 \leq i_0 < \ldots < i_p \leq c$ and $1 \leq j_0 < \ldots < j_q \leq c$ we denote $U_{i_0 \ldots i_p} \subset X$, $V_{j_0 \ldots j_q} \subset X$, and $W_{i_0 \ldots i_p, j_0 \ldots j_q} \subset X$ the open subscheme where $f_{i_0} \ldots f_{i_p}$ is invertible, $g_{j_0} \ldots g_{j_q}$ is invertible, and where $f_{i_0} \ldots f_{i_p}g_{j_0} \ldots g_{j_q}$ is invertible. We denote $\mathcal{F}_{i_0 \ldots i_p}$, resp.\ $\mathcal{F}'_{j_0 \ldots j_q}$ $\mathcal{F}''_{i_0 \ldots i_p, j_0 \ldots j_q}$ the pushforward to $X$ of the restriction of $\mathcal{F}$ to $U_{i_0 \ldots i_p}$, resp.\ $V_{j_0 \ldots j_q}$, resp.\ $W_{i_0 \ldots i_p, j_0 \ldots j_q}$. Then we obtain three extended alternating {\v C}ech complexes $$ \mathcal{F}^\bullet : \mathcal{F} \to \bigoplus\nolimits_{i_0} \mathcal{F}_{i_0} \to \bigoplus\nolimits_{i_0 < i_1} \mathcal{F}_{i_0i_1} \to \ldots $$ and $$ (\mathcal{F}')^\bullet : \mathcal{F} \to \bigoplus\nolimits_{j_0} \mathcal{F}'_{j_0} \to \bigoplus\nolimits_{j_0 < j_1} \mathcal{F}'_{j_0j_1} \to \ldots $$ and $$ (\mathcal{F}'')^\bullet : \mathcal{F} \to \bigoplus\nolimits_{i_0} \mathcal{F}_{i_0} \oplus \bigoplus\nolimits_{j_0} \mathcal{F}'_{j_0} \to \bigoplus\nolimits_{i_0 < i_1} \mathcal{F}_{i_0i_1} \oplus \bigoplus\nolimits_{i_0, j_0} \mathcal{F}''_{i_0, j_0} \oplus \bigoplus\nolimits_{j_0 < j_1} \mathcal{F}'_{j_0j_1} \to \ldots $$ whose differentials are those used in defining (\ref{equation-extended-alternating}). There are maps of complexes $$ (\mathcal{F}'')^\bullet \to \mathcal{F}^\bullet \quad\text{and}\quad (\mathcal{F}'')^\bullet \to (\mathcal{F}')^\bullet $$ given by the projection maps on the terms (and hence inducing the identity map in degree $0$). Observe that by Lemma \ref{lemma-extended-alternating-represented} each of these complexes represents $i_*R\mathcal{H}_Z(\mathcal{F})$ and these maps represent the identity on this object. Thus it suffices to find an element $$ \sigma \in H^c((\mathcal{F}'')^\bullet(X)) $$ mapping to $c_{f_1, \ldots, f_c}(s)$ and $\det(a_{ji})c_{g_1, \ldots, g_c}(s)$ by these two maps. It turns out we can explicitly give a cocycle for $\sigma$. Namely, we take $$ \sigma_{1 \ldots c} = \frac{s}{f_1 \ldots f_c} \in \mathcal{F}_{1 \ldots c}(X) \quad\text{and}\quad \sigma'_{1 \ldots c} = \frac{\det(a_{ji})s}{g_1 \ldots g_c} \in \mathcal{F}'_{1 \ldots c}(X) $$ and we take $$ \sigma_{i_0 \ldots i_p, j_0 \ldots j_{c - p - 2}} = \frac{\lambda(i_0 \ldots i_p, j_0 \ldots j_{c - p - 2})s}% {f_{i_0} \ldots f_{i_p}g_{j_0} \ldots g_{j_{c - p - 2}}} \in \mathcal{F}''_{i_0 \ldots i_p, j_0 \ldots j_{c - p - 2}}(X) $$ where $\lambda(i_0 \ldots i_p, j_0 \ldots j_{c - p - 2})$ is the coefficient of $e_1 \wedge \ldots \wedge e_c$ in the formal expresssion $$ e_{i_0} \wedge \ldots \wedge e_{i_p} \wedge (a_{j_01} e_1 + \ldots + a_{j_0c}e_c) \wedge \ldots \wedge (a_{j_{c - p - 2}1} e_1 + \ldots + a_{j_{c - p - 2}c}e_c) $$ To verify that $\sigma$ is a cocycle, we have to show for $1 \leq i_0 < \ldots < i_p \leq c$ and $1 \leq j_0 < \ldots < j_{c - p - 1} \leq c$ that we have \begin{align*} 0 & = \sum\nolimits_{a = 0, \ldots, p} (-1)^a f_{i_a} \lambda(i_0 \ldots \hat i_a \ldots i_p, j_0 \ldots j_{c - p - 1}) \\ & + \sum\nolimits_{b = 0, \ldots, c - p - 1} (-1)^{p + b + 1}g_{j_b} \lambda(i_0 \ldots i_p, j_0 \ldots \hat j_b \ldots j_{c - p - 1}) \end{align*} The easiest way to see this is perhaps to argue that the formal expression $$ \xi = e_{i_0} \wedge \ldots \wedge e_{i_p} \wedge (a_{j_01} e_1 + \ldots + a_{j_0c}e_c) \wedge \ldots \wedge (a_{j_{c - p - 1}1} e_1 + \ldots + a_{j_{c - p - 1}c}e_c) $$ is $0$ as it is an element of the $(c + 1)$st wedge power of the free module on $e_1, \ldots, e_c$ and that the expression above is the image of $\xi$ under the Koszul differential sending $e_i \to f_i$. Some details omitted.