Let $F\colon \mathcal A \to \mathcal B$ be a left exact functor between Grothendieck abelian categories. Given a morphism $f\colon A\to B$ in $\mathcal A$ and a morphism $g\colon RF(A)\to RF(B)$ in the derived category of $\mathcal B$, is there a “standard” technique to determine whether $g=RF(f)$? Take “standard” to mean what you will.
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$\begingroup$ Good question, especially because verifications like these are sometimes swept under the rug. But $g$ being given in a somewhat canonical fashion does not guarantee that actually $g = RF(f)$; in particular, this might hold only up to sign. In computer science, there is something called "theorems for free" which is probably relevant to this question. $\endgroup$ – Ingo Blechschmidt Dec 14 '18 at 22:37