This is a combination of the results mentioned in the statement. The adjointness can be shown as follows: Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_\mathcal{X}$-module and let $\mathcal{G}$ be a quasi-coherent $\mathcal{O}_\mathcal{Y}$-module. Then we have \begin{align*} \Mor_{\QCoh(\mathcal{O}_\mathcal{X})}(f^*\mathcal{G}, \mathcal{F}) & = \Mor_{\mathcal{M}_\mathcal{Y}}(\mathcal{G}, f_*\mathcal{F}) \\ & = \Mor_{\QCoh(\mathcal{O}_\mathcal{Y})}(\mathcal{G}, Q(f_*\mathcal{F})) \\ & = \Mor_{\QCoh(\mathcal{O}_\mathcal{Y})}(\mathcal{G}, f_{\QCoh, *}\mathcal{F}) \end{align*} the first equality by adjointness of $f_*$ and $f^*$ (for arbitrary sheaves of modules). By Proposition \ref{proposition-lcq-flat-base-change} we see that $f_*\mathcal{F}$ is an object of $\mathcal{M}_\mathcal{Y}$ (and can be computed in either the fppf or \'etale topology) and we obtain the second equality by Lemma \ref{lemma-adjoint}. The third equality is the definition of $f_{\QCoh, *}$. \medskip\noindent To see that $\{R^if_{\QCoh, *}\}_{i \geq 0}$ is a cohomological $\delta$-functor as defined in Homology, Definition \ref{homology-definition-cohomological-delta-functor} let $$ 0 \to \mathcal{F}_1 \to \mathcal{F}_2 \to \mathcal{F}_3 \to 0 $$ be a short exact sequence of $\QCoh(\mathcal{O}_\mathcal{X})$. This sequence may not be an exact sequence in $\textit{Mod}(\mathcal{O}_\mathcal{X})$ but we know that it is up to parasitic modules, see Lemma \ref{lemma-exact-sequence-quasi-coherent-parasitic-cohomology}. Thus we may break up the sequence into short exact sequences $$ \begin{matrix} 0 \to \mathcal{P}_1 \to \mathcal{F}_1 \to \mathcal{I}_2 \to 0 \\ 0 \to \mathcal{I}_2 \to \mathcal{F}_2 \to \mathcal{Q}_2 \to 0 \\ 0 \to \mathcal{P}_2 \to \mathcal{Q}_2 \to \mathcal{I}_3 \to 0 \\ 0 \to \mathcal{I}_3 \to \mathcal{F}_3 \to \mathcal{P}_3 \to 0 \end{matrix} $$ of $\textit{Mod}(\mathcal{O}_\mathcal{X})$ with $\mathcal{P}_i$ parasitic. Note that each of the sheaves $\mathcal{P}_j$, $\mathcal{I}_j$, $\mathcal{Q}_j$ is an object of $\mathcal{M}_\mathcal{X}$, see Proposition \ref{proposition-lcq-flat-base-change}. Applying $R^if_*$ we obtain long exact sequences  $$ \begin{matrix} 0 \to f_*\mathcal{P}_1 \to f_*\mathcal{F}_1 \to f_*\mathcal{I}_2 \to R^1f_*\mathcal{P}_1 \to \ldots \\ 0 \to f_*\mathcal{I}_2 \to f_*\mathcal{F}_2 \to f_*\mathcal{Q}_2 \to R^1f_*\mathcal{I}_2 \to \ldots \\ 0 \to f_*\mathcal{P}_2 \to f_*\mathcal{Q}_2 \to f_*\mathcal{I}_3 \to R^1f_*\mathcal{P}_2 \to \ldots \\ 0 \to f_*\mathcal{I}_3 \to f_*\mathcal{F}_3 \to f_*\mathcal{P}_3 \to R^1f_*\mathcal{I}_3 \to \ldots \end{matrix} $$ where are the terms are objects of $\mathcal{M}_\mathcal{Y}$ by Proposition \ref{proposition-lcq-flat-base-change}. By Lemma \ref{lemma-pushforward-parasitic} the sheaves $R^if_*\mathcal{P}_j$ are parasitic, hence vanish on applying the functor $Q$, see Lemma \ref{lemma-adjoint-kernel-parasitic}. Since $Q$ is exact the maps $$ Q(R^if_*\mathcal{F}_3) \cong Q(R^if_*\mathcal{I}_3) \cong Q(R^if_*\mathcal{Q}_2) \rightarrow Q(R^{i + 1}f_*\mathcal{I}_2) \cong Q(R^{i + 1}f_*\mathcal{F}_1) $$ can serve as the connecting map which turns the family of functors $\{R^if_{\QCoh, *}\}_{i \geq 0}$ into a cohomological $\delta$-functor.