A balanced derived bifunctor

Assume the Axiom of Dependent Choice. Let $\mathcal{A},\mathcal{C},\mathcal{D}$ be abelian categories, let $B:{\mathcal{A}}^{\mathrm{o}\mathrm{p}}\times \mathcal{C}\to \mathcal{D}$ be additive in each variable, let $P$ be supplied projective resolution data on a class ${\mathcal{D}}_A$ in $\mathcal{A}$, and let $I$ be supplied injective resolution data on a class ${\mathcal{D}}_C$ in $\mathcal{C}$. Assume moreover that for each fixed $A\in {\mathcal{D}}_A$ the covariant functor $B(A,-):\mathcal{C}\to \mathcal{D}$ is left exact, and that for each fixed $C\in {\mathcal{D}}_C$ the functor $B(-,C):{\mathcal{A}}^{\mathrm{o}\mathrm{p}}\to \mathcal{D}$ is left exact.

A balanced derived bifunctor relative to $(P,I)$ on ${\mathcal{D}}_A^{\mathrm{o}\mathrm{p}}\times {\mathcal{D}}_C$ consists of the two candidate one-variable right-derived constructions from A bifunctor can be derived in either variable when the relevant resolution data are supplied together with, for every $n\ge 0$, a natural isomorphism $R_I^n(B(A,-))(C)\stackrel{\sim }{\to }{R}_{P^{\mathrm{o}\mathrm{p}}}^n(B(-,C))(A)$ natural in $A\in {\mathcal{D}}_A$ and $C\in {\mathcal{D}}_C$. These isomorphisms must satisfy:

1. in degree $0$, when the two candidates are identified with $B(A,C)$ by The zero-th right derived functor of a left exact functor recovers the functor on $\mathcal{C}$ and on ${\mathcal{A}}^{\mathrm{o}\mathrm{p}}$, the balance isomorphism becomes the identity of $B(A,C)$;
2. the isomorphisms are natural in both variables in the sense of Natural transformation and its components.

This is a definition relative to the displayed supplied data $P$ and $I$; it does not impose an unquantified condition involving alternative data. The definition records extra comparison data, while the previous proposition only constructs the two candidates.