For any multi-index $(i_0, \ldots, i_p) \in I^{p + 1}$ there exists a unique permutation $\sigma : \{0, \ldots, p\} \to \{0, \ldots, p\}$ such that $$ i_{\sigma(0)} \leq i_{\sigma(1)} \leq \ldots \leq i_{\sigma(p)} \quad \text{and} \quad \sigma(j) < \sigma(j + 1) \quad \text{if} \quad i_{\sigma(j)} = i_{\sigma(j + 1)}. $$ We denote this permutation $\sigma = \sigma^{i_0 \ldots i_p}$. \medskip\noindent For any permutation $\sigma : \{0, \ldots, p\} \to \{0, \ldots, p\}$ and any $a$, $0 \leq a \leq p$ we denote $\sigma_a$ the permutation of $\{0, \ldots, p\}$ such that $$ \sigma_a(j) = \left\{ \begin{matrix} \sigma(j) & \text{if} & 0 \leq j < a, \\ \min\{j' \mid j' > \sigma_a(j - 1), j' \not = \sigma(k), \forall k < a\} & \text{if} & a \leq j \end{matrix} \right. $$ So if $p = 3$ and $\sigma$, $\tau$ are given by $$ \begin{matrix} \text{id} & 0 & 1 & 2 & 3 \\ \sigma & 3 & 2 & 1 & 0 \end{matrix} \quad \text{and} \quad \begin{matrix} \text{id} & 0 & 1 & 2 & 3 \\ \tau & 3 & 0 & 2 & 1 \end{matrix} $$ then we have $$ \begin{matrix} \text{id} & 0 & 1 & 2 & 3 \\ \sigma_0 & 0 & 1 & 2 & 3 \\ \sigma_1 & 3 & 0 & 1 & 2 \\ \sigma_2 & 3 & 2 & 0 & 1 \\ \sigma_3 & 3 & 2 & 1 & 0 \\ \end{matrix} \quad \text{and} \quad \begin{matrix} \text{id} & 0 & 1 & 2 & 3 \\ \tau_0 & 0 & 1 & 2 & 3 \\ \tau_1 & 3 & 0 & 1 & 2 \\ \tau_2 & 3 & 0 & 1 & 2 \\ \tau_3 & 3 & 0 & 2 & 1 \\ \end{matrix} $$ It is clear that always $\sigma_0 = \text{id}$ and $\sigma_p = \sigma$. \medskip\noindent Having introduced this notation we define for $s \in \check{\mathcal{C}}^{p + 1}(\mathcal{U}, \mathcal{F})$ the element $h(s) \in \check{\mathcal{C}}^p(\mathcal{U}, \mathcal{F})$ to be the element with components \begin{equation} \label{equation-first-homotopy} h(s)_{i_0\ldots i_p} = \sum\nolimits_{0 \leq a \leq p} (-1)^a \text{sign}(\sigma_a) s_{i_{\sigma(0)} \ldots i_{\sigma(a)} i_{\sigma_a(a)} \ldots i_{\sigma_a(p)}} \end{equation} where $\sigma = \sigma^{i_0 \ldots i_p}$. The index $i_{\sigma(a)}$ occurs twice in $i_{\sigma(0)} \ldots i_{\sigma(a)} i_{\sigma_a(a)} \ldots i_{\sigma_a(p)}$ once in the first group of $a + 1$ indices and once in the second group of $p - a + 1$ indices since $\sigma_a(j) = \sigma(a)$ for some $j \geq a$ by definition of $\sigma_a$. Hence the sum makes sense since each of the elements $s_{i_{\sigma(0)} \ldots i_{\sigma(a)} i_{\sigma_a(a)} \ldots i_{\sigma_a(p)}}$ is defined over the open $U_{i_0 \ldots i_p}$. Note also that for $a = 0$ we get $s_{i_0 \ldots i_p}$ and for $a = p$ we get $(-1)^p \text{sign}(\sigma) s_{i_{\sigma(0)} \ldots i_{\sigma(p)}}$. \medskip\noindent We claim that $$ (dh + hd)(s)_{i_0 \ldots i_p} = s_{i_0 \ldots i_p} - \text{sign}(\sigma) s_{i_{\sigma(0)} \ldots i_{\sigma(p)}} $$ where $\sigma = \sigma^{i_0 \ldots i_p}$. We omit the verification of this claim. (There is a PARI/gp script called first-homotopy.gp in the stacks-project subdirectory scripts which can be used to check finitely many instances of this claim. We wrote this script to make sure the signs are correct.) Write $$ \kappa : \check{\mathcal{C}}^\bullet(\mathcal{U}, \mathcal{F}) \longrightarrow \check{\mathcal{C}}^\bullet(\mathcal{U}, \mathcal{F}) $$ for the operator given by the rule $$ \kappa(s)_{i_0 \ldots i_p} = \text{sign}(\sigma^{i_0 \ldots i_p}) s_{i_{\sigma(0)} \ldots i_{\sigma(p)}}. $$ The claim above implies that $\kappa$ is a morphism of complexes and that $\kappa$ is homotopic to the identity map of the {\v C}ech complex. This does not immediately imply the lemma since the image of the operator $\kappa$ is not the alternating subcomplex. Namely, the image of $\kappa$ is the ``semi-alternating'' complex $\check{\mathcal{C}}_{semi\text{-}alt}^p(\mathcal{U}, \mathcal{F})$ where $s$ is a $p$-cochain of this complex if and only if $$ s_{i_0 \ldots i_p} = \text{sign}(\sigma) s_{i_{\sigma(0)} \ldots i_{\sigma(p)}} $$ for any $(i_0, \ldots, i_p) \in I^{p + 1}$ with $\sigma = \sigma^{i_0 \ldots i_p}$. We introduce yet another variant {\v C}ech complex, namely the semi-ordered {\v C}ech complex defined by $$ \check{\mathcal{C}}_{semi\text{-}ord}^p(\mathcal{U}, \mathcal{F}) = \prod\nolimits_{i_0 \leq i_1 \leq \ldots \leq i_p} \mathcal{F}(U_{i_0 \ldots i_p}) $$ It is easy to see that Equation (\ref{equation-d-cech}) also defines a differential and hence that we get a complex. It is also clear (analogous to Lemma \ref{lemma-project-to-ordered}) that the projection map $$ \check{\mathcal{C}}_{semi\text{-}alt}^\bullet(\mathcal{U}, \mathcal{F}) \longrightarrow \check{\mathcal{C}}_{semi\text{-}ord}^\bullet(\mathcal{U}, \mathcal{F}) $$ is an isomorphism of complexes. \medskip\noindent Hence the Lemma follows if we can show that the obvious inclusion map $$ \check{\mathcal{C}}_{ord}^p(\mathcal{U}, \mathcal{F}) \longrightarrow \check{\mathcal{C}}_{semi\text{-}ord}^p(\mathcal{U}, \mathcal{F}) $$ is a homotopy equivalence. To see this we use the homotopy \begin{equation} \label{equation-second-homotopy} h(s)_{i_0 \ldots i_p} = \left\{ \begin{matrix} 0 & \text{if} & i_0 < i_1 < \ldots < i_p \\ (-1)^a s_{i_0 \ldots i_{a - 1} i_a i_a i_{a + 1} \ldots i_p} & \text{if} & i_0 < i_1 < \ldots < i_{a - 1} < i_a = i_{a + 1} \end{matrix} \right. \end{equation} We claim that $$ (dh + hd)(s)_{i_0 \ldots i_p} = \left\{ \begin{matrix} 0 & \text{if} & i_0 < i_1 < \ldots < i_p \\ s_{i_0 \ldots i_p} & \text{else} & \end{matrix} \right. $$ We omit the verification. (There is a PARI/gp script called second-homotopy.gp in the stacks-project subdirectory scripts which can be used to check finitely many instances of this claim. We wrote this script to make sure the signs are correct.) The claim clearly shows that the composition $$ \check{\mathcal{C}}_{semi\text{-}ord}^\bullet(\mathcal{U}, \mathcal{F}) \longrightarrow \check{\mathcal{C}}_{ord}^\bullet(\mathcal{U}, \mathcal{F}) \longrightarrow \check{\mathcal{C}}_{semi\text{-}ord}^\bullet(\mathcal{U}, \mathcal{F}) $$ of the projection with the natural inclusion is homotopic to the identity map as desired.