I'm totally stuck with this problem that I found in an Algebra course. It is the following:
Let $F:\mathcal{A} \to \mathcal{B}$ be a left exact functor between two abelian cathegories. Let $\mathcal{F}$ be a family of objects of $\mathcal{A}$ with these two properties:
(i) For every object $x$ of $\mathcal{A}$ there is $y\in \mathcal{F}$ and a monomorphism $\phi:x\to y.$
(ii) for every short exact sequence $0 \to a \to b \to c \to 0$ with $a,b \in \mathcal{F},$ we have the properties $c \in \mathcal{F}$ and $ 0 \to Fa \to Fb \to Fc \to 0$ is exact.
Then for every complex bounded from below $a^*$ with $a^n\in \mathcal{F}$ for every $n$ we have a quasi-isomorphism $F(a^*)\to RF(a^*).$
$RF(a)$ is defined as follows: take the complex $0\to a^0\to a^1\to ...$, take an injective resolution $0\to i^0\to i^1\to...$ and consider the complex $RF(a^*)=F(i^*)$.
I showed that the functor $RF$ is well defined (up to homotopic equivalence) and sends distinguished triangles to distinguished triangles.
Using property (ii) I showed that $F$ preserves exactness for long exact sequences of the form $0\to b^0\to b^1\to...$ with $b^n\in\mathcal{F}$.
I still have no idea about how to use (i): maybe I may try to construct resolutions made of objects of $\mathcal{F}$? Would these be somehow related to injective resolutions? Sadly I didn't come out with anything. Thanks to everybody.