Assume all the categories here are abelian and the functors are additive. Given functors $F: \mathcal{C} \rightarrow \mathcal{D}$ and $G : \mathcal{D} \rightarrow \mathcal{D}$, we say that they are an adjoint pair if there is a natural bijection $$ \text{Hom}(F(a), b) \simeq \text{Hom}(a, G(b)). $$ Is there a term for when there is a natural bijection of Ext groups, $$ \text{Ext}^{i}(F(a), b) \simeq \text{Ext}^{i}(a, G(b))? $$ Further, since $F$ may fail to be left exact and $G$ may fail to be right exact, is there a term for when the derived functors are adjoint?
This has come up in a few topics I am studying, but results are generally just stated as known. I'm interested at looking at this in the general case, but I don't know what the terminology is to look for. Is there a standard reference that covers this, or even just a name for these functors?
