The Stacks project

Remark 24.33.3. As above, let $\mathcal{C}$ be a category viewed as a site with the chaotic topology, let $\mathcal{O}$ be a sheaf of rings on $\mathcal{C}$, and let $(\mathcal{A}, \text{d})$ be a sheaf of differential graded $\mathcal{O}$-algebras. Then the analogue of Cohomology on Sites, Proposition 21.43.9 holds for $\mathit{QC}(\mathcal{A}, \text{d})$ with almost exactly the same proof:

1. any contravariant cohomological functor $H : \mathit{QC}(\mathcal{A}, \text{d}) \to \textit{Ab}$ which transforms direct sums into products is representable,

2. any exact functor $F : \mathit{QC}(\mathcal{A}, \text{d}) \to \mathcal{D}$ of triangulated categories which transforms direct sums into direct sums has an exact right adjoint, and

3. the inclusion functor $\mathit{QC}(\mathcal{A}, \text{d}) \to D(\mathcal{A}, \text{d})$ has an exact right adjoint.

If we ever need this we will precisely formulate and prove this here.