It's confusing to use $E$ to denote both an element of $\operatorname{Ext}^1$ and an object of $\mathcal {A}$.

It's not true in general that $F$ preserves quasi-isomorphisms between unbounded complexes of objects acyclic for $F$, or even between unbounded complexes of injectives.

Do you mean to write "Right derived functor" in the title? Isn't the point that you're applying the functor $F$ rather than its right derived functor, which is exact and so always preserves quasi-isomorphisms?

Also, it's not known that every derived equivalence is induced by a two-sided tilting complex (i.e., is "standard" in the terminology of the same paper).

I'm not sure exactly what you mean by "perform this construction". If you mean precisely the construction used in the proof of Corollary 5.5 in that paper, then the answer is no, but if you allow variations of that construction, then maybe.

@varkor Well, it's not the same for every categorical concept. E.g., monomorphism/epimorphism. But even for those that use the "co-" construction, the better ones have a root that indicates which way the arrows point: e.g., kernel/cokernel or well-powered/co-well-powered. I think limit/colimit is a particularly bad case, because not only did somebody make an arbitrary choice, but I would argue that they made the wrong arbitrary choice!

@varkor I know it’s a common convention (or dually, a mmon nvention), but I think it’s a bad (and ugly) one. I think that mathematical terminology should be chosen to be descriptive, and if you start with “extremal monomorphism”, dualize it to “coextremal epimorphism”, and then detach the adjectives to apply them to more general morphisms, what happens? You end up with two terms “extremal morphism” and “coextremal morphism”, where which is which depends on what is pretty much the historical accident of whether “extremal” was first applied to monomorphisms or epimorphisms.

@varkor I disagree that extremal epimorphisms should be called "coextremal". They are extremal in exactly the same way that extremal monomorphisms are. They don't have the dual property to being extremal. The Arctic is in the extreme North of the planet, but the Antarctic is not in the coextreme South; it's in the extreme South.

@Bumblebee Sorry, no, I don't think I've seen it before. But I'd call it an observation rather than a theorem.