Let $\mathcal{A}$ and $\mathcal{B}$ be two abelian categories and let $\mathcal{F}:\mathcal{A}\to \mathcal{B}$ be an additive functor. Assume that $\mathcal{F}$ is exact and let $\mathcal{D}(\mathcal{F}):\mathcal{D}(\mathcal{A})\to \mathcal{D}(\mathcal{B})$ denotes its (right and left) derived functor.

Under which assumptions on $\mathcal{F}$ is $\mathcal{D}(\mathcal{F})$ fully faithful? (Surely, $\mathcal{F}$ would be fully faithful, but is it enough?)

Many thanks!