The following
DERIVED YONEDA LEMMA
is well known: let $\mathcal{A}$ an abelian category, $T:\mathcal{A}\rightarrow\mathbf{Ab}$ a right exact functor, and $S$ a fixed object of $\mathcal{A}$. Then we have a canonical bijection $$Hom(Ext^n(S,-),F)\rightarrow (L_n F)(S).$$
Are there generalizations of this result phrased on the level at derived categories? I would also be interested in applications of this derived Yoneda lemma.