The Stacks project

Remark 56.5.8. Let $R$ be a ring. Let $X$ and $Y$ be schemes over $R$. Assume $X$ is quasi-compact with affine diagonal. Lemma 56.5.7 may be generalized as follows: the functors (56.5.1.1) associated to quasi-coherent modules on $X \times _ R Y$ are exactly those $F : \mathit{QCoh}(\mathcal{O}_ X) \to \mathit{QCoh}(\mathcal{O}_ Y)$ which have the following properties

1. $F$ is $R$-linear and commutes with arbitrary direct sums,

2. $F \circ j_*$ is right exact when $j : U \to X$ is the inclusion of an affine open, and

3. $0 \to F(\mathcal{F}) \to F(\mathcal{G}) \to F(\mathcal{H})$ is exact whenever $0 \to \mathcal{F} \to \mathcal{G} \to \mathcal{H} \to 0$ is an exact sequence such that for all $x \in X$ the sequence on stalks $0 \to \mathcal{F}_ x \to \mathcal{G}_ x \to \mathcal{H}_ x \to 0$ is a split short exact sequence.

Namely, these assumptions are enough to get construct a transformation $t : F \to F_\mathcal {K}$ as in Lemma 56.5.5 and to show that it is an isomorphism. Moreover, properties (1), (2), and (3) do hold for functors (56.5.1.1). If we ever need this we will carefully state and prove this here.