Since $\epsilon^{-1}\mathcal{O}_{\tau'} = \mathcal{O}_\tau$ we see that $\epsilon$ is a flat morphism of ringed sites and that in fact $\epsilon^{-1} = \epsilon^*$ on sheaves of modules. By property (1) we can think of every object of $\mathcal{A}$ as a sheaf of $\mathcal{O}_\tau$-modules and as a sheaf of $\mathcal{O}_{\tau'}$-modules. In other words, we have fully faithful inclusion functors $$ \mathcal{A} \to \textit{Mod}(\mathcal{O}_\tau) \to \textit{Mod}(\mathcal{O}_{\tau'}) $$ To avoid confusion we will denote $\mathcal{A}' \subset \textit{Mod}(\mathcal{O}_{\tau'})$ the image of $\mathcal{A}$. Then it is clear that $\epsilon_* : \mathcal{A} \to \mathcal{A}'$ and $\epsilon^* : \mathcal{A}' \to \mathcal{A}$ are quasi-inverse equivalences (see discussion preceding the lemma and use that objects of $\mathcal{A}'$ are sheaves in the $\tau$ topology). \medskip\noindent Conditions (3) and (4) imply that $R^p\epsilon_*\mathcal{F} = 0$ for $p > 0$ and $\mathcal{F} \in \Ob(\mathcal{A})$. This is true because $R^p\epsilon_*$ is the sheaf associated to the presheave $U \mapsto H^p_\tau(U, \mathcal{F})$, see Lemma \ref{lemma-higher-direct-images}. Thus any exact complex in $\mathcal{A}$ (which is the same thing as an exact complex in $\textit{Mod}(\mathcal{O}_\tau)$ whose terms are in $\mathcal{A}$, see Homology, Lemma \ref{homology-lemma-characterize-weak-serre-subcategory}) remains exact upon applying the functor $\epsilon_*$. \medskip\noindent Consider an exact sequence $$ \mathcal{F}'_0 \to \mathcal{F}'_1 \to \mathcal{F}'_2 \to \mathcal{F}'_3 \to \mathcal{F}'_4 $$ in $\textit{Mod}(\mathcal{O}_{\tau'})$ with $\mathcal{F}'_0, \mathcal{F}'_1, \mathcal{F}'_3, \mathcal{F}'_4$ in $\mathcal{A}'$. Apply the exact functor $\epsilon^*$ to get an exact sequence $$ \epsilon^*\mathcal{F}'_0 \to \epsilon^*\mathcal{F}'_1 \to \epsilon^*\mathcal{F}'_2 \to \epsilon^*\mathcal{F}'_3 \to \epsilon^*\mathcal{F}'_4 $$ in $\textit{Mod}(\mathcal{O}_\tau)$. Since $\mathcal{A}$ is a weak Serre subcategory and since $\epsilon^*\mathcal{F}'_0, \epsilon^*\mathcal{F}'_1, \epsilon^*\mathcal{F}'_3, \epsilon^*\mathcal{F}'_4$ are in $\mathcal{A}$, we conclude that $\epsilon^*\mathcal{F}_2$ is in $\mathcal{A}$ by Homology, Definition \ref{homology-definition-serre-subcategory}. Consider the map of sequences $$ \xymatrix{ \mathcal{F}'_0 \ar[r] \ar[d] & \mathcal{F}'_1 \ar[r] \ar[d] & \mathcal{F}'_2 \ar[r] \ar[d] & \mathcal{F}'_3 \ar[r] \ar[d] & \mathcal{F}'_4 \ar[d] \\ \epsilon_*\epsilon^*\mathcal{F}'_0 \ar[r] & \epsilon_*\epsilon^*\mathcal{F}'_1 \ar[r] & \epsilon_*\epsilon^*\mathcal{F}'_2 \ar[r] & \epsilon_*\epsilon^*\mathcal{F}'_3 \ar[r] & \epsilon_*\epsilon^*\mathcal{F}'_4 } $$ The lower row is exact by the discussion in the preceding paragraph. The vertical arrows with index $0$, $1$, $3$, $4$ are isomorphisms by the discussion in the first paragraph. By the $5$ lemma (Homology, Lemma \ref{homology-lemma-five-lemma}) we find that $\mathcal{F}'_2 \cong \epsilon_*\epsilon^*\mathcal{F}'_2$ and hence $\mathcal{F}'_2$ is in $\mathcal{A}'$. In this way we see that $\mathcal{A}'$ is a weak Serre subcategory of $\textit{Mod}(\mathcal{O}_{\tau'})$, see Homology, Definition \ref{homology-definition-serre-subcategory}. \medskip\noindent At this point it makes sense to talk about the derived categories $D_\mathcal{A}(\mathcal{O}_\tau)$ and $D_{\mathcal{A}'}(\mathcal{O}_{\tau'})$, see Derived Categories, Section \ref{derived-section-triangulated-sub}. To finish the proof we show that conditions (1) -- (5) of Lemma \ref{lemma-equivalence-unbounded-two} apply. We have already seen (1), (2), (3) above. Note that since every object has a $\tau'$-covering by objects of $\mathcal{B}$, a fortiori every object has a $\tau$-covering by objects of $\mathcal{B}$. Hence condition (4) of Lemma \ref{lemma-equivalence-unbounded-two} is satisfied. Similarly, condition (5) is satisfied as well.