The Stacks project

Proof. Write $E_ n = E|_{X_ n}$ and $M_ n = R\Gamma (X_ n, K_ n \otimes ^\mathbf {L} E|_{X_ n})$. By Derived Categories of Spaces, Lemma 75.25.5 we see that $M_ n$ is a pseudo-coherent object of $D(A_ n)$ whose formation commutes with base change. Thus the maps $M_ n \otimes _{A_ n}^\mathbf {L} A_{n - 1} \to M_{n - 1}$ coming from $\varphi _ n$ are isomorphisms. By More on Algebra, Lemma 15.97.1 we find that $R\mathop{\mathrm{lim}}\nolimits M_ n$ is pseudo-coherent and that its base change back to $A_ n$ recovers $M_ n$. On the other hand, the exact functor $R\Gamma (X, -) : D_\mathit{QCoh}(\mathcal{O}_ X) \to D(A)$ of triangulated categories commutes with products and hence with derived limits, whence

as desired. $\square$