An F-acyclic resolution

Let $F$ be an additive functor between abelian categories.

1. If $F$ is left exact and $I$ is a supplied injective resolution datum on a class $\mathcal{D}$, an $F$-acyclic resolution of $A$ relative to $I$ is a coaugmented exact complex $0\to A\to J^0\to J^1\to \cdots$ such that every $J^q$ lies in $\mathcal{D}$ and is $F$-acyclic in the sense of An acyclic object for a left exact functor.
2. If $F$ is right exact and $P$ is a supplied projective resolution datum on a class $\mathcal{D}$, an $F$-acyclic resolution of $A$ relative to $P$ is an augmented exact complex $\cdots \to P_1\to P_0\to A\to 0$ such that every $P_q$ lies in $\mathcal{D}$ and is $F$-acyclic in the sense of An acyclic object for a right exact functor.

Thus the phrase keeps both the resolution orientation and the chosen supplied datum visible.