(Apologies if this question is trivial, but I'm way outside my area here.)

Let $R$ be a commutative ring, $C^{\bullet}(R)$ the category of complexes of $R$-modules, and $D^{\bullet}(R)$ its derived category. Suppose we have an object $X \in \operatorname{Obj} D^{\bullet}(R)$, and an endomorphism $t \in \operatorname{End}_{D^{\bullet}(R)} X$. By definition, this just means that we can write $t$ as a formal fraction $s^{-1} f$, where $f: X^\bullet\to Y^\bullet$ is a map of complexes and $s: X^\bullet \to Y^\bullet$ is a quasi-isomorphism.

Can we always arrange that $X^\bullet = Y^\bullet$? That is, do endomorphisms in the derived category always lift to endomorphisms of some (or even any) representing complex?

Something of this kind seems to be used at the bottom of page 10 of this paper by Khare and Thorne, and I'm trying to work out why this is true.