Let $p: X\rightarrow S$ be a map between topological spaces and we can construct the simplicial space $\text{cosq}(X\rightarrow S)$ where $X_0=X$, $X_1=X\times_S X$ and $$ X_n=\underbrace{ X\times_S \ldots\times_S X}_{n+1 \text{ components}} $$ with face maps just deleting certain components and degeneracy maps certain diagonal maps.
Let's look at the bounded derived category of abelian groups $D^b(\text{cosq}(X\rightarrow S))$ on $\text{cosq}(X\rightarrow S)$. An object $\mathcal{F}_{\cdot}$is a collection $\mathcal{F}_n \in obj(D^b(X_n))$ and for any simplicial map $h: X_n\rightarrow X_m$ we have the structure morphism $$ \alpha_h: h^*(\mathcal{F}_m)\rightarrow \mathcal{F}_m $$ which satisfies the compatible condition $\alpha_{h^{\prime}h}=\alpha_h\circ h^*\alpha_{h^{\prime}}$.
A morphism $\phi: \mathcal{F}_{\cdot}\rightarrow \mathcal{G}_{\cdot}$ in $D^b(\text{cosq} (X\rightarrow S))$ is a collection of morphisms $\phi_n: \mathcal{F}_n\rightarrow \mathcal{G}_n$ which are compatible with the structure morphisms.
We have a full subcategory $ D^b_{\text{Car}}(\text{cosq}(X\rightarrow S))$ of $D^b(\text{cosq} (X\rightarrow S))$ with objects the $\mathcal{F}_{\cdot}$'s such that all structure morphisms $\alpha_h: h^*(\mathcal{F}_m)\rightarrow \mathcal{F}_m$ are quasi-isomorphisms.
It is easy to check that the from $p: X\rightarrow S$ we get the pull-back map $p^*$ $$ p^*: D^b(S)\rightarrow D^b_{\text{Car}}(\text{cosq}(X\rightarrow S)). $$
Now in Zhiwei Yun's Notes on Equivariant Derived Categories there is a general result:
$\textbf{Propostion 6}$ in the above notes: If $p: X\rightarrow S$ locally admits a section, then the pull back map $$ p^*: D^b(S)\rightarrow D^b_{\text{Car}}(\text{cosq}(X\rightarrow S)). $$ is an equivalence of triangulated categories.
I think this result is well-known to experts $\textbf{My question}$ is how can we proof it?