We will use the commutative diagram (\ref{equation-commutative-epsilon}) without further mention. In particular have $$ Rf_{\tau', *}R\epsilon_{X, *}\mathcal{F} = R\epsilon_{Y, *}Rf_{\tau, *}\mathcal{F} $$ Assumption $(V_n)$ tells us that $\epsilon_{X, *}\mathcal{F} \to R\epsilon_{X, *}\mathcal{F}$ is an isomorphism in degrees $\leq n$. Hence $Rf_{\tau', *}\epsilon_{X, *}\mathcal{F} \to Rf_{\tau', *}R\epsilon_{X, *}\mathcal{F}$ is an isomorphism in degrees $\leq n$. We conclude that $$ R^if_{\tau', *}\epsilon_{X, *}\mathcal{F} \to H^i(R\epsilon_{Y, *}Rf_{\tau, *}\mathcal{F}) $$ is an isomorphism for $i \leq n$. We will prove the lemma by looking at the second page of the spectral sequence of Lemma \ref{lemma-relative-Leray} for $R\epsilon_{Y, *}Rf_{\tau, *}\mathcal{F}$. Here is a picture: $$ \begin{matrix} \ldots & \ldots & \ldots & \ldots \\ \epsilon_{Y, *}R^2f_{\tau, *}\mathcal{F} & R^1\epsilon_{Y, *}R^2f_{\tau, *}\mathcal{F} & R^2\epsilon_{Y, *}R^2f_{\tau, *}\mathcal{F} & \ldots \\ \epsilon_{Y, *}R^1f_{\tau, *}\mathcal{F} & R^1\epsilon_{Y, *}R^1f_{\tau, *}\mathcal{F} & R^2\epsilon_{Y, *}R^1f_{\tau, *}\mathcal{F} & \ldots \\ \epsilon_{Y, *}f_{\tau, *}\mathcal{F} & R^1\epsilon_{Y, *}f_{\tau, *}\mathcal{F} & R^2\epsilon_{Y, *}f_{\tau, *}\mathcal{F} & \ldots \end{matrix} $$ Let $(C_m)$ be the hypothesis:  $R^if_{\tau', *}\epsilon_{X, *}\mathcal{F} = \epsilon_{Y, *}R^if_{\tau, *}\mathcal{F}$ for $i \leq m$. Observe that $(C_0)$ holds. We will show that $(C_{m - 1}) \Rightarrow (C_m)$ for $m < n$. Namely, if $(C_{m - 1})$ holds, then for $n \geq p > 0$ and $q \leq m - 1$ we have \begin{align*} R^p\epsilon_{Y, *}R^qf_{\tau, *}\mathcal{F} & = R^p\epsilon_{Y, *} \epsilon_Y^{-1} \epsilon_{Y, *} R^qf_{\tau, *}\mathcal{F} \\ & = R^p\epsilon_{Y, *} \epsilon_Y^{-1}R^qf_{\tau', *}\epsilon_{X, *}\mathcal{F} = 0 \end{align*} First equality as $\epsilon_Y^{-1}\epsilon_{Y, *} = \text{id}$, the second by $(C_{m - 1})$, and the final by by $(V_n)$ because $\epsilon_Y^{-1}R^qf_{\tau', *}\epsilon_{X, *}\mathcal{F}$ is in $\mathcal{A}_Y$ by (\ref{item-A-and-P}). Looking at the spectral sequence we see that $E_2^{0, m} = \epsilon_{Y, *}R^mf_{\tau, *}\mathcal{F}$ is the only nonzero term $E_2^{p, q}$ with $p + q = m$. Recall that $\text{d}_r^{p, q} : E_r^{p, q} \to E_r^{p + r, q - r + 1}$. Hence there are no nonzero differentials $\text{d}_r^{p, q}$, $r \geq 2$ either emanating or entering this spot. We conclude that $H^m(R\epsilon_{Y, *}Rf_{\tau, *}\mathcal{F}) = \epsilon_{Y, *}R^mf_{\tau, *}\mathcal{F}$ which implies $(C_m)$ by the discussion above. \medskip\noindent Finally, assume $(C_{n - 1})$. The same analysis shows that $E_2^{0, n} = \epsilon_{Y, *}R^nf_{\tau, *}\mathcal{F}$ is the only nonzero term $E_2^{p, q}$ with $p + q = n$. We do still have no nonzero differentials entering this spot, but there can be a nonzero differential emanating it. Namely, the map $d_{n + 1}^{0, n} : \epsilon_{Y, *}R^nf_{\tau, *}\mathcal{F} \to R^{n + 1}\epsilon_{Y, *}f_{\tau, *}\mathcal{F}$. We conclude that there is an exact sequence $$ 0 \to R^nf_{\tau', *}\epsilon_{X, *}\mathcal{F} \to  \epsilon_{Y, *}R^nf_{\tau, *}\mathcal{F} \to R^{n + 1}\epsilon_{Y, *}f_{\tau, *}\mathcal{F} $$ By (\ref{item-A-and-P}) and (\ref{item-A-sheaf}) the sheaf $R^nf_{\tau', *}\epsilon_{X, *}\mathcal{F}$ satisfies the sheaf property for $\tau$-coverings as does $\epsilon_{Y, *}R^nf_{\tau, *}\mathcal{F}$ (use the description of $\epsilon_*$ in Section \ref{section-compare}). However, the $\tau$-sheafification of the $\tau'$-sheaf $R^{n + 1}\epsilon_{Y, *}f_{\tau, *}\mathcal{F}$ is zero (by locality of cohomology; use Lemmas \ref{lemma-kill-cohomology-class-on-covering} and \ref{lemma-higher-direct-images}). Thus $R^nf_{\tau', *}\epsilon_{X, *}\mathcal{F} \to  \epsilon_{Y, *}R^nf_{\tau, *}\mathcal{F}$ has to be an isomorphism and the proof is complete.