The Stacks project

Suggested slogan: Derived functors are compatible with shifts

There is a slight asymmetry between the statement of (1) and (2). The former uses $=$ and the latter $\to$.

$\def\colim#1{\underset{#1}{\operatorname{colim}}}$(Lest this comment not render properly in your browser, here is the original code.)

A proof of (1):

We have $$\begin{align*} \left(\colim{(X\to X')\in X/S}F(X')\right)[n] &\cong\colim{(X\to X')\in X/S}F(X')[n]\\ &\cong\colim{(X\to X')\in X/S}F(X'[n])\\ &\cong\colim{(X[n]\to X')\in X[n]/S}F(X'), \end{align*}$$ where all these colimits exist in $\mathcal{D}'$ if and only if one of them does, and where the last isomorphism comes from application of Categories, Lemma 4.17.2 to the cofinal functor $[n]:X/S\to X[n]/S$ (well-defined thanks to MS5; it is cofinal for it is an equivalence). The first term of these chain of isos is $RF(X)[n]$, whereas the last one is $RF(X[n])$.