G-acyclic object for a left-exact functor

Let $\mathcal{B}$ and $\mathcal{C}$ be abelian categories, let $G:\mathcal{B}\to \mathcal{C}$ be additive and left exact, and let $X\in \mathcal{B}$. Fix a supplied injective resolution datum $I$ whose domain contains $X$. The object $X$ is $G$-acyclic relative to $I$ if $R_I^{p}G(X)=0$ for every integer $p>0$, where $R_I^{p}G(X)$ is defined by Right derived objects relative to supplied injective resolution data. A complex is termwise $G$-acyclic relative to supplied data if each term $X^n$ lies in the domain of a specified datum $I_n$ and satisfies $R_{I_n}^{p}G(X^n)=0$ for every $p>0$.

The condition imposes no vanishing on $G(X)$, which is canonically isomorphic to $R_I^{0}G(X)$ by left exactness and the resolution augmentation; it does not assert literal equality of these objects. It is distinct from exactness of a complex: exactness concerns its differentials, whereas termwise $G$-acyclicity concerns the higher derived objects of its individual terms. The zero object with its zero resolution is $G$-acyclic relative to that datum; a complex with no terms satisfies termwise acyclicity vacuously. No choice of a family of resolutions or independence assertion is part of this definition.