We are going to show that $H^p(U, f^*\mathcal{F}) = H^p_\tau(U, \mathcal{F}^a)$ for any object $f : U \to S$ of the site $\mathcal{C}$. The result is true for $p = 0$ by the sheaf property. \medskip\noindent Assume that $U$ is affine. Then we want to prove that $H^p_\tau(U, \mathcal{F}^a) = 0$ for all $p > 0$. We use induction on $p$. \begin{enumerate} \item[p = 1] Pick $\xi \in H^1_\tau(U, \mathcal{F}^a)$. By Lemma \ref{lemma-locality-cohomology}, there exists an fpqc covering $\mathcal{U} = \{U_i \to U\}_{i \in I}$ such that $\xi|_{U_i} = 0$ for all $i \in I$. Up to refining $\mathcal{U}$, we may assume that $\mathcal{U}$ is a standard $\tau$-covering. Applying the spectral sequence of Theorem \ref{theorem-cech-ss}, we see that $\xi$ comes from a cohomology class $\check \xi \in \check H^1(\mathcal{U}, \mathcal{F}^a)$. Consider the covering $\mathcal{V} = \{\coprod_{i\in I} U_i \to U\}$. By Lemma \ref{lemma-cech-complex}, $\check H^\bullet(\mathcal{U}, \mathcal{F}^a) = \check H^\bullet(\mathcal{V}, \mathcal{F}^a)$. On the other hand, since $\mathcal{V}$ is a covering of the form $\{\Spec(B) \to \Spec(A)\}$ and $f^*\mathcal{F} = \widetilde{M}$ for some $A$-module $M$, we see the {\v C}ech complex $\check{\mathcal{C}}^\bullet(\mathcal{V}, \mathcal{F})$ is none other than the complex $(B/A)_\bullet \otimes_A M$. Now by Lemma \ref{lemma-descent-modules}, $H^p((B/A)_\bullet \otimes_A M) = 0$ for $p > 0$, hence $\check \xi = 0$ and so $\xi = 0$. \item[p > 1] Pick $\xi \in H^p_\tau(U, \mathcal{F}^a)$. By Lemma \ref{lemma-locality-cohomology}, there exists an fpqc covering $\mathcal{U} = \{U_i \to U\}_{i \in I}$ such that $\xi|_{U_i} = 0$ for all $i \in I$. Up to refining $\mathcal{U}$, we may assume that $\mathcal{U}$ is a standard $\tau$-covering. We apply the spectral sequence of Theorem \ref{theorem-cech-ss}. Observe that the intersections $U_{i_0} \times_U \ldots \times_U U_{i_p}$ are affine, so that by induction hypothesis the cohomology groups $$ E_2^{p, q} = \check H^p(\mathcal{U}, \underline{H}^q(\mathcal{F}^a)) $$ vanish for all $0 < q < p$. We see that $\xi$ must come from a $\check \xi \in \check H^p(\mathcal{U}, \mathcal{F}^a)$. Replacing $\mathcal{U}$ with the covering $\mathcal{V}$ containing only one morphism and using Lemma \ref{lemma-descent-modules} again, we see that the {\v C}ech cohomology class $\check \xi$ must be zero, hence $\xi = 0$. \end{enumerate} Next, assume that $U$ is separated. Choose an affine open covering $U = \bigcup_{i \in I} U_i$ of $U$. The family $\mathcal{U} = \{U_i \to U\}_{i \in I}$ is then an fpqc covering, and all the intersections $U_{i_0} \times_U \ldots \times_U U_{i_p}$ are affine since $U$ is separated. So all rows of the spectral sequence of Theorem \ref{theorem-cech-ss} are zero, except the zeroth row. Therefore $$ H^p_\tau(U, \mathcal{F}^a) = \check H^p(\mathcal{U}, \mathcal{F}^a) = \check H^p(\mathcal{U}, \mathcal{F}) = H^p(U, \mathcal{F}) $$ where the last equality results from standard scheme theory, see Cohomology of Schemes, Lemma \ref{coherent-lemma-cech-cohomology-quasi-coherent}. \medskip\noindent The general case is technical and (to extend the proof as given here) requires a discussion about maps of spectral sequences, so we won't treat it. It follows from Descent, Proposition \ref{descent-proposition-same-cohomology-quasi-coherent} (whose proof takes a slightly different approach) combined with Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-cohomology-of-open}.