Let $\mathcal{K}$ be as in (2). The functor $F$ in (\ref{equation-FM-QCoh}) commutes with direct sums. Since by (1) (a) the modules $\mathcal{K}$ is $X$-flat, we see that given a short exact sequence $0 \to \mathcal{F}_1 \to \mathcal{F}_2 \to \mathcal{F}_3 \to 0$ we obtain a short exact sequence $$ 0 \to \text{pr}_1^*\mathcal{F}_1 \otimes_{\mathcal{O}_{X \times_R Y}} \mathcal{K} \to \text{pr}_1^*\mathcal{F}_2 \otimes_{\mathcal{O}_{X \times_R Y}} \mathcal{K} \to \text{pr}_1^*\mathcal{F}_3 \otimes_{\mathcal{O}_{X \times_R Y}} \mathcal{K} \to 0 $$ Since by (2)(b) the higher direct image $R^1\text{pr}_{2, *}$ on the first term is zero, we conclude that $0 \to F(\mathcal{F}_1) \to F(\mathcal{F}_2) \to F(\mathcal{F}_3) \to 0$ and we see that $F$ is as in (1). \medskip\noindent Let us construct the quasi-inverse to the construction. Let $F$ be as in (1). Choose an affine open covering $X = \bigcup_{i = 1, \ldots, n} U_i$. Since the diagonal of $X$ is affine, we see that the intersections $U_{i_0 \ldots i_p} = U_{i_0} \cap \ldots \cap U_{i_p}$ are affine and that the inclusion morphisms $j_{i_0 \ldots i_p} : U_{i_0 \ldots i_p} \to X$ are affine. See Morphisms, Lemma \ref{morphisms-lemma-affine-permanence}. In particular, the composition $$ \QCoh(\mathcal{O}_{U_{i_0 \ldots i_p}}) \xrightarrow{j_{i_0 \ldots i_p *}} \QCoh(\mathcal{O}_X) \xrightarrow{F} \QCoh(\mathcal{O}_Y) $$ is an exact functor commuting with direct sums as a composition of such functors. By Lemmas \ref{lemma-functor-quasi-coherent-from-affine} and \ref{lemma-coh-noetherian-from-affine-flat} this functor is given by a quasi-coherent module $\mathcal{K}_{i_0 \ldots i_p}$ on $U_{i_0 \ldots i_p} \times_R Y$ flat over $U_{i_0 \ldots i_p}$. Since $$ \QCoh(\mathcal{O}_{U_{i_0 \ldots i_p i_{p + 1}}}) \xrightarrow{(U_{i_0 \ldots i_p i_{p + 1}} \to U_{i_0 \ldots i_p})_*} \QCoh(\mathcal{O}_{U_{i_0 \ldots i_p}}) \xrightarrow{j_{i_0 \ldots i_p *}} \QCoh(\mathcal{O}_X) $$ is equal to $j_{i_0 \ldots i_p i_{p + 1} *}$ we conclude from Lemma \ref{lemma-functor-quasi-coherent-from-affine-compose} and the equivalence of categories of the already used Lemma \ref{lemma-functor-quasi-coherent-from-affine} that we obtain identifications $$ \mathcal{K}_{i_0 \ldots i_p i_{p + 1}} = \mathcal{K}_{i_0 \ldots i_p}|_{U_{i_0 \ldots i_p i_{p + 1}} \times_R Y} $$ which satisfy the usual compatibilites for glueing. In other words, there exists a unique $\mathcal{K} \in \QCoh(\mathcal{O}_{X \times_R Y})$ flat over $X$ which restricts to each $\mathcal{K}_{i_0 \ldots i_p}$ on $U_{i_0 \ldots i_p} \times_R Y$ compatible with these identifications. For every quasi-coherent $\mathcal{O}_X$-module we have the sheafified {\v C}ech complex $$ 0 \to \mathcal{F} \to \bigoplus\nolimits_{i_0} j_{i_0 *}\mathcal{F}|_{U_{i_0}} \to \bigoplus\nolimits_{i_0i_1} j_{i_0 i_1 *}\mathcal{F}|_{U_{i_0 i_1}} \to \ldots $$ which is exact. See Cohomology, Lemma \ref{cohomology-lemma-covering-resolution}. Applying the exact functor $F$ we find that $F(\mathcal{F})$ maps quasi-isomorphically to the relative {\v C}ech complex with terms $$ \bigoplus\nolimits_{i_0 \ldots i_p} (U_{i_0 \ldots i_p} \times_R Y \to Y)_*( \text{pr}_1^*\mathcal{F} \otimes_{\mathcal{O}_{X \times_R Y}} \mathcal{K} )|_{U_{i_0 \ldots i_p} \times_R Y} $$ Since this {\v C}ech complex computes the pushfoward and higher direct images of $\text{pr}_1^*\mathcal{F} \otimes_{\mathcal{O}_{X \times_R Y}} \mathcal{K}$ by $\text{pr}_2$ (by Cohomology of Schemes, Lemma \ref{coherent-lemma-separated-case-relative-cech}) we conclude $F$ and $\mathcal{K}$ correspond and that we have property (2)(b).