In the Stacks project Tag 08TC, there is a definition of a homotopy limit in a derived category, and I expect it to compare with a limit in the $\infty$-categorical enhancement. I guess this is also hinted at in the beginning of the proof of Proposition 1.2.1.19 in Lurie's Higher Algebra. However, I have trouble to write down the details. Could someone clarify this for me please? Also, is there a reference available?
Alternatively, is there a way to see this through the Bousfield-Kan formula?
